Choice numbers of graphs

TEL-A VIV UNIVERSITY RA YMOND AND BEVERL Y SACKLER F A CUL TY OF EXACT SCIENCES Choice n um b ers of graphs This work w as subm it ted in partial fulfillment of the requirement s f o r th e Maste r’s degree (M.Sc.) at T el-Aviv Unive r s it y Sc ho ol of Mathematical Sciences Departmen t of Computer S c ience Presen ted by Shai Gutner This work w as carried ou t u nder the sup ervision of Prof. Mic hael T arsi July 1992 I am gr ateful to my advisor, Pr of. Michael T arsi, for his guidanc e and assistanc e thr oughout the work on this thesis. I would like to thank Pr of. No ga Alon for helpful discussions and c omments. 2 Con ten ts 1 In tro duction 4 2 A solution to a problem of Erd˝ os, Rubin and T ay lor 8 3 An upp er b ound for t he k th c hoice num b er 9 4 Choice n umbers and orientations 13 5 Prop erties of (2 k : k ) -c hoosable graphs 16 6 The complexit y of graph c ho osabilit y 24 7 The strong choic e num b er 27 3 1 In tro duction A graph G = ( V , E ) is ( a : b ) -cho osable if for ev ery family of sets { S ( v ) : v ∈ V } , where | S ( v ) | = a for all v ∈ V , ther e are subsets C ( v ) ⊆ S ( v ), where | C ( v ) | = b f or all v ∈ V , and C ( u ) ∩ C ( v ) = ∅ for ev ery t wo adjacen t v ertices u, v ∈ V . The k th choic e numb er of G , denoted by ch k ( G ), is the minim um in teger n so that G is ( n : k )-c ho osa b le . A graph G = ( V , E ) is k -cho osable if it is ( k : 1)-c ho osable. The choic e numb er of G , denoted by ch ( G ), is equal to ch 1 ( G ). The concept of ( a : b )-c ho osabilit y was defined and studied b y Erd˝ os, Rubin and T a ylor in [8]. In the p r ese nt p a p er w e pro v e s everal results concerning ( a : b )-choosabilit y , a n umber of wh ic h generalize kno wn r esults regarding choice num b ers of graphs that app ear in [4] and [2]. The follo wing theorem examines th e b eha vior of ch k ( G ) wh en k is large. Theorem 1.1 L et G b e a g r aph. F or every ǫ > 0 ther e exists an inte ger k 0 such that ch k ( G ) ≤ k ( χ ( G ) + ǫ ) for every k ≥ k 0 . In [8] the authors ask the f o llo win g qu e stion: If G is ( a : b )-c ho osa ble, and c d > a b , do es it follo w that G is ( c : d )-c ho osable? The follo wing corollary giv es a n egativ e answer to this question. Corollary 1.2 If l > m ≥ 3 , then ther e is a gr aph G which is ( a : b ) - c h o osable but not ( c : d ) - cho osable wher e c d = l and a b = m . Let K m ∗ r denote the co mp le te r -partite graph w it h m v ertices in eac h v ertex class, and let K m 1 ,...,m r denote the complete r -partite graph with m i v ertices in the i th ve rtex class. It is sho wn in [2] that there exist t wo p ositiv e constan ts c 1 and c 2 suc h t h at for eve ry m ≥ 2 and for ev ery r ≥ 2, c 1 r log m ≤ ch ( K m ∗ r ) ≤ c 2 r log m . The follo wing theorem generalizes the upp er b ound. Theorem 1.3 If r ≥ 1 and m i ≥ 2 for every i , 1 ≤ i ≤ r , then ch k ( K m 1 ,...,m r ) ≤ 948 r ( k + log m 1 + · · · + m r r ) . The follo wing are t wo applications of this theorem. Corollary 1.4 F or every gr aph G and k ≥ 1 ch k ( G ) ≤ 948 χ ( G ) ( k + log ( | V | χ ( G ) + 1) ) . 4 The second corollary generalizes a result from [2] concerning the choic e num b ers of random graphs for the common mo del G n,p (see, e.g., [7]), in wh ic h the graph is obtained by taking eac h pair of the n lab eled v ertices 1 , 2 , . . . , n to b e an edge, rand o mly and in d epend en tly , with pr o bability p . Corollary 1.5 F or every two c onstants k ≥ 1 and 0 < p < 1 , the pr ob ability that ch k ( G n,p ) ≤ 475 log (1 / (1 − p )) n log log n log n tends to 1 as n tends to infinity. A th eorem wh ic h app ears in [4] rev eals the connection b et w een the c hoice num b er of a graph G and its orienta tions. W e pr e sent here a generalization of th is th eorem for a sp ecial case. Theorem 1.6 L et D = ( V , E ) b e a digr aph and k ≥ 1 . F or e ach v ∈ V , let S ( v ) b e a set of size k ( d + D ( v ) + 1) , wher e d + D ( v ) is the outde g r e e of v. If D c ontains no o dd dir e cte d (simple) cycle, then ther e ar e subsets C ( v ) ⊆ S ( v ) , wher e | C ( v ) | = k for al l v ∈ V , and C ( u ) ∩ C ( v ) = ∅ for every two adjac ent vertic es u, v ∈ V . Ther e is a p olyno mial time algorithm in | V | and k which finds the subsets C ( v ) . Corollary 1.7 L et G b e an undir e cte d gr aph . If G has a n orienta tion D which c ontains no o dd dir e cte d (simple) cycle in which the maximum outde gr e e is d, then G is ( k ( d + 1) : k ) -cho osable for every k ≥ 1 . Corollary 1.8 An even c ycle is (2 k : k ) -cho osable for every k ≥ 1 . The last corollary enables us to p r o v e a generalization of a v arian t of Brooks T heorem w hic h app ears in [8]. Corollary 1.9 If a c onne cte d gr aph G is not K n , and not an o dd cycle, then ch k ( G ) ≤ k ∆( G ) for every k ≥ 1 , wher e ∆( G ) is the maximum de gr e e of G . F or a graph G = ( V , E ), defi n e M ( G ) = max( | E ( H ) | / | V ( H ) | ), where H = ( V ( H ) , E ( H )) ranges o v er all subgraphs of G . The follo wing t wo corollaries are generalizations of results wh ic h app ear in [4]. Corollary 1.10 Every b ip artite gr aph G is ( k ( ⌈ M ( G ) ⌉ + 1) : k ) -cho osable for al l k ≥ 1 . Corollary 1.11 Every b ip artite planar gr aph G is (3 k : k ) -cho osa ble for al l k ≥ 1 . 5 The follo wing are additional applications. Corollary 1.12 If ev e r y induc e d sub gr aph of a gr aph G has a vertex of de gr e e at most d , then G is ( k ( d + 1) : k ) -cho osable for al l k ≥ 1 . Corollary 1.13 If G is a triangulate d gr ap h, then ch k ( G ) = kχ ( G ) = k ω ( G ) for every k ≥ 1 , wher e ω ( G ) is the clique numb er of G . The list-c hromatic conjecture asserts that for ev ery graph G , ch ( L ( G )) = χ ( L ( G )), where L ( G ) is the line graph of G . The list-c hromatic conjecture is easy to establish for trees, graphs of degree at m o st 2, and K 2 ,m . It has also b een verified for snarks [11], K 3 , 3 , K 4 , 4 , K 6 , 6 [4], and 2-connected cubic p la n a r graphs . Th e follo wing co rollary sh o ws that the list-c hromatic conjecture is true for graphs which conta in no C n for ev ery n ≥ 4. Corollary 1.14 If a gr aph G c ontains no C n for every n ≥ 4 , then ch ( L ( G )) = χ ( L ( G )) . The c or e of a graph G is the graph obtained f r om G by deleting no des of degree 1 successive ly unt il there are no no des of degree 1. The graph Θ a,b,c consists of tw o d istin gu ish ed no des u an d v together with three p a ths of lengths a , b , and c , which are no de disjoin t except that eac h path has u at one end , and v at the other end . The follo wing theorem from [8] giv es a c haracterization of the 2-c ho osable graph: Theorem 1.15 A c on ne cte d gr aph G is 2 -cho osable if, and only if, the c or e of G b elongs to { K 1 , C 2 m +2 , Θ 2 , 2 , 2 m : m ≥ 1 } . In [8] the authors ask the f o llo win g qu e stion: If G is ( a : b )-c ho osa ble, do es it follo w that G is ( am : bm )-c ho osable? The follo wing theorem giv es a partial solution to this qu est ion by using theorem 1.15. Theorem 1.16 If a gr aph G is 2 -cho osable, then G is also (4 : 2) -cho osable. Theorem 1.17 Supp ose that k and m ar e p ositive inte gers and that k is o dd. If a gr aph G is (2 mk : mk ) -cho osa ble, then G is also 2 m -cho osable. A graph G = ( V , E ) is f -cho osable for a function f : V 7→ N if for every f a mily of sets { S ( v ) : v ∈ V } , w here | S ( v ) | = f ( v ) for all v ∈ V , there is a pr op er verte x-coloring of G assigning 6 to eac h v ertex v ∈ V a color from S ( v ). It is shown in [8] that the foll owing problem is Π p 2 -complete: ( for terminology see [10] ) BIP AR TITE GRAPH (2 , 3) -CHOOSABILITY ( B G (2 , 3) -CH) INST ANCE: A b ip a rtite graph G = ( V , E ) and a fun ct ion f : V 7→ { 2 , 3 } . QUESTION: Is G f -c ho o sable? W e consider the follo win g d ecision problem: BIP AR TITE GRAPH k -CHOO SAB ILI TY (BG k -CH) INST ANCE: A b ip a rtite graph G = ( V , E ). QUESTION: Is G k -c ho osable? If follo ws from theorem 1.15 that this pr o blem is solv ab le in p olynomial time for k = 2. Theorem 1.18 BIP AR TIT E GRAPH k -CHOOSABILI TY is Π p 2 -c omplete for every c on- stant k ≥ 3 . A graph G = ( V , E ) is str ongly k -c olor able if ev ery grap h obtained from G by addin g to it a union of v ertex disjoin t cliques of size at most k ( on the set V ) is k -colorable. An a nalogous definition of str ongly k -cho osa ble is made by replacing colorabilit y with choosabilit y . T he str ong chr omatic numb er of a graph G , denoted by sχ ( G ), is the minimum k suc h that G is strongly k -co lorable. Defin e sχ ( d ) = max( sχ ( G )), where G ranges o v er all graphs with maximum d e gree at most d . The defin it ion of strongly k -colorable giv en in [1] is sligh tly different. I t is claimed ther e that if G is strongly k -colo rable, then it is strongly ( k + 1)-colorable as well . Ho w ev er, it is not kno wn h o w to pro ve this if we u s e the defi nitio n f rom [1]. Theorem 1.19 If G is str ongly k -c olor able, then it is str ongly ( k + 1) -c olor able as wel l. W e giv e a w eak er version of this theorem for c ho osabilit y . Theorem 1.20 If G is str ongly k -cho osable, then it is str ongly k m -cho osable as wel l. Theorem 1.21 L et G = ( V , E ) b e a gr aph , and supp ose that k m divides | V | . If the choic e numb er of any gr aph obtaine d fr om G by adding to it a union of vertex disjoint k -cliques (on the se t V ) is k , then the choic e numb er of any gr ap h obtaine d fr om G by adding to it a union of v e r tex disjoint k m -cliques is k m . 7 Corollary 1.22 L et n and k b e p ositive i nte gers, and let G b e a (3 k + 1) -r e gular gr aph on 3 k n vertic es. A ssume that G has a de c omp osition into a Hamiltonian cir cuit and n p airwise vertex disjoint 3 k -cliques. Then ch ( G ) = 3 k . It is p ro v ed in [1] th a t there is a constan t c such that for ev ery d , 3 ⌊ d/ 2 ⌋ < sχ ( d ) ≤ cd . The follo wing theorem impr ov es the lo w er b ound . Theorem 1.23 F or every d ≥ 1 , s χ ( d ) ≥ 2 d . 2 A solution to a problem of Erd˝ os, Rub i n and T a ylor In this section w e pro ve an upp er b ound for the k th c hoice num b er of a graph w hen k is large and apply this b ound to settle a problem raised in [8]. Pro of of Theorem 1.1 Let G = ( V , E ) b e a graph and ǫ > 0. Denote r = χ ( G ), and let V = V 1 ∪ . . . ∪ V r b e a partition of the v ertices, s uc h that eac h V i is a stable set. F or eac h v ∈ V , let S ( v ) b e a set of ⌊ k ( χ ( G ) + ǫ ) ⌋ distinct colors. Let S = ∪ v ∈ V S ( v ) b e the set of all colors. Pu t R = { 1 , 2 , . . . , r } and let f : S 7→ R b e a random function, obtained b y c ho osing, for eac h color c ∈ S , rand o mly and ind epend e ntly , the v alue of f ( c ) according to a u niform distribution on R . The colo rs c for whic h f ( c ) = i will b e the ones to b e used for co loring the v ertices in V i . T o complete the pro of, it thus su ffice s to sho w that with p ositiv e probabilit y for eve ry i , 1 ≤ i ≤ r , and for eve ry ve rtex v ∈ V i there are at least k colors c ∈ S ( v ) so that f ( c ) = i . Fix an i and a v ertex v ∈ V i , and defin e X = | S ( v ) ∩ f − 1 ( i ) | . The p r obabilit y that there are less than k colors c ∈ S ( v ) so that f ( c ) = i is equal to P r ( X < k ). Since X is a rand om v ariable with distribu ti on B ( ⌊ k ( r + ǫ ) ⌋ , 1 /r ), by Ch eb yshev’s inequalit y (see, e.g., [3]) P r ( X < k ) ≤ P r ( | X − ⌊ k ( r + ǫ ) ⌋ r | ≥ ⌊ k ǫ ⌋ r ) ≤ ⌊ k ( r + ǫ ) ⌋ 1 r (1 − 1 r ) ( ⌊ kǫ ⌋ r ) 2 = O ( 1 k ) . It follo ws that th e re is an int eger k 0 suc h that P ( X < k ) < 1 / | V | for every k ≥ k 0 . Ther e are | V | p ossible c h oices of i , 1 ≤ i ≤ r and v ∈ V i , and hence, the probab ility that for some i and s o me v ∈ V i there are less than k colors c ∈ S ( v ) so th at f ( c ) = i is sm aller than 1, completing th e pr oof. ✷ Note that it is not true that for ev ery graph G there exists an int eger k 0 suc h that ch k ( G ) ≤ k χ ( G ) for every k ≥ k 0 . F or example, tak e the graph G = K 3 , 3 whic h has c hromatic n umb er 2. 8 The graph G is not 2-c ho osable and therefore by theorem 1.17 it is not (2 k : k )-c ho osable for ev ery k o dd. This means that ch k ( G ) > k χ ( G ) for ev ery k o dd. Pro of of Corollary 1.2 S upp ose that l > m ≥ 3, and let G b e a graph s u c h that ch ( G ) = l + 1 and χ ( G ) = m − 1 ( it is pro ved in [13] that for ev ery l ≥ m ≥ 2 there is a graph G , where ch ( G ) = l and χ ( G ) = m ). By theorem 1.1 , for ǫ = 1 there exist an in teger k su c h that G is ( k ( χ ( G ) + 1) : k )-c ho osable. W e ha ve that G is ( k m : k )-c ho osable but not ( l : 1)-c ho osable, as needed. ✷ 3 An upp er b oun d for the k th c hoice n um b er In this section we establish an upp er b ound for ch k ( K m 1 ,...,m r ), and u se it to pro v e t wo consequences. The follo wing lemma app ears in [3]. Lemma 3.1 If X is a r and om variable with distribution B ( n, p ) , 0 < p ≤ 1 , and k < pn then P r ( X < k ) < e − ( np − k ) 2 2 pn . In the rest of this section w e d enot e t = m 1 + ··· + m r r , t 1 = m 1 + ··· + m r / 2 r / 2 , and t 2 = m r / 2 +1 + ··· + m r r / 2 . Notice that t = ( t 1 + t 2 ) / 2, and therefore log t 1 t 2 ≤ 2 log t . Lemma 3.2 If 1 ≤ r ≤ t , k ≥ 1 , and m i ≥ 2 for every i , 1 ≤ i ≤ r , then ch k ( K m 1 ,...,m r ) ≤ 4 r ( k + log t ) . Pro of Let V 1 , V 2 , . . . , V r b e the vertex classes of K = K m 1 ,...,m r , w here | V i | = m i for all i , and let V = V 1 ∪ . . . ∪ V r b e th e set of all vertice s of K . F or eac h v ∈ V , let S ( v ) b e a set of ⌊ 4 r ( k + log t ) ⌋ distinct colors. Put R = { 1 , 2 , . . . , r } and let f : S 7→ R b e a rand om function, obtained by c ho osing, for eac h color c ∈ S , rand o mly and indep endently , the v alue of f ( c ) according to a uniform distribu ti on on R . Th e colors c f o r w hic h f ( c ) = i will b e the ones to b e used for coloring the ve rtices in V i . T o complete the pro of it thus su ffices to sh ow that with p ositiv e probab ility f o r ev ery i , 1 ≤ i ≤ r , an d ev ery v ertex v ∈ V i there are at least k colors c ∈ S ( v ) so that f ( c ) = i . Fix an i and a v ertex v ∈ V i , and defin e X = | S ( v ) ∩ f − 1 ( i ) | . The p r obabilit y that there are less than k colors c ∈ S ( v ) so that f ( c ) = i is equal to P r ( X < k ). Since X is a rand om v ariable with distribu ti on B ( ⌊ 4 r ( k + log t ) ⌋ , 1 /r ), by lemma 3.1 P r ( X < k ) < e − ( E ( X ) − k ) 2 2 E ( X ) ≤ e − (4( k +log t ) − 1 − k ) 2 8( k +log t ) < e − 16( k +log t ) 2 − 8( k +1)( k + log t ) 8( k +log t ) ≤ e − 2 log t = 1 t 2 ≤ 1 r t , 9 where the last inequalit y follo ws from the fact that r ≤ t . Th ere are r t p ossible c h oices of i , 1 ≤ i ≤ r and v ∈ V i , and hence, the probab ility that for some i and s o me v ∈ V i there are less than k colors c ∈ S ( v ) so that f ( c ) = i is smaller th a n 1, completing th e pro of. ✷ Lemma 3.3 Supp ose that r is ev en, r > t , k ≥ 1 , d ≥ 244 , and m i ≥ 2 for e v ery i , 1 ≤ i ≤ r . If ch k ( K m 1 ,...,m r / 2 ) ≤ d (1 − 1 5 r 1 / 3 ) r 2 ( k + log t 1 ) and ch k ( K m r / 2 +1 ,...,m r ) ≤ d (1 − 1 5 r 1 / 3 ) r 2 ( k + log t 2 ) , then ch k ( K m 1 ,...,m r ) ≤ dr ( k + log t ) . Pro of Let V 1 , V 2 , . . . , V r b e the vertex classes of K = K m 1 ,...,m r , w here | V i | = m i for all i , and let V = V 1 ∪ . . . ∪ V r b e the set of all ve rtices of K . F or eac h v ∈ V , let S ( v ) b e a set of ⌊ dr ( k + log t ) ⌋ distinct colors. Define R = { 1 , 2 , . . . , r } , and let S = ∪ v ∈ V S ( v ) b e the s et of all colors. Put R 1 = { 1 , 2 , . . . , r / 2 } and R 2 = { r / 2 + 1 , . . . , r } . Let f : S 7→ { 1 , 2 } b e a random function obtained b y choosing, for eac h c ∈ S randomly and ind epend en tly , f ( c ) ∈ { 1 , 2 } w here for all j ∈ { 1 , 2 } P r ( f ( c ) = j ) = k + log t j 2 k + log t 1 t 2 . The colors c for whic h f ( c ) = 1 will b e used for coloring the ve rtices in ∪ i ∈ R 1 V i , whereas the colors c for wh ic h f ( c ) = 2 will b e used for coloring the vertice s in ∪ i ∈ R 2 V i . F or every v ertex v ∈ V , defin e C ( v ) = S ( v ) ∩ f − 1 (1) if v b elongs to ∪ i ∈ R 1 V i , and C ( v ) = S ( v ) ∩ f − 1 (2) if v b elongs to ∪ i ∈ R 2 V i . Beca us e of the assump ti ons of the lemma, it remains to sho w that with p ositiv e p r obabilit y , | C ( v ) | ≥ d (1 − 1 5 r 1 / 3 ) r 2 ( k + log t j ) (1) for all j ∈ { 1 , 2 } and v ∈ ∪ i ∈ R j V i . Fix a j ∈ { 1 , 2 } and a vertex v ∈ ∪ i ∈ R j V i , and d efine X = | C ( v ) | . Th e exp ectatio n of X is ⌊ dr ( k + log t ) ⌋ k + log t j 2 k + log t 1 t 2 ≥ ( dr ( k + log t ) − 1) k + log t j 2 k + 2 log t ≥ d r 2 ( k + log t j ) − 1 = T . If follo ws from lemma 3.1 and the inequalit y E ( X ) ≥ T th a t P r ( X < T − T 2 / 3 ) < e − ( E ( X ) − T + T 2 / 3 ) 2 2 E ( X ) ≤ e − 1 2 T 1 / 3 ≤ e − 1 2 ( d r 2 ) 1 / 3 . Since | ∪ i ∈ R j V i | ≤ r t < r 2 , the p r obabilit y that | C ( v ) | < T − T 2 / 3 holds for some v ∈ ∪ i ∈ R j V i is at most r 2 · e − 1 2 ( d r 2 ) 1 / 3 < 1 / 2 , 10 where the last inequalit y follo ws f rom the fact that d ≥ 244. One can easily c hec k that T − T 2 / 3 = T (1 − 1 T 1 / 3 ) ≥ d r 2 ( k + log t j )(1 − 1 5 r 1 / 3 ) , and therefore, w ith p ositiv e probabilit y (1) h o lds for all j ∈ { 1 , 2 } and v ∈ ∪ i ∈ R j V i . ✷ Pro of of Theorem 1.3 Define for ev ery r wh ic h is a p o w er of 2 f ( r ) = log 2 r Y j =0 (1 − 1 5 · 2 j / 3 ) / 2 Y j =0 (1 − 1 5 · 2 j / 3 ) . W e claim that f o r ev ery r which is a p o w er of 2 ch k ( K m 1 ,...,m r ) ≤ 244 r ( k + log t ) f ( r ) . (2) The pro of is by indu ct ion on r . Case 1: r ≤ t . The result follo ws fr o m lemma 3.2 since 244 f ( r ) ≥ 244 2 Y j =1 (1 − 1 5 · 2 j / 3 ) > 4 . Case 2: r > t . Notice that t ≥ 2, and therefore r ≥ 4. By the indu c tion hyp ot hesis ch k ( K m 1 ,...,m r / 2 ) ≤ 244(1 − 1 5 r 1 / 3 ) r 2 ( k + log t 1 ) f ( r ) and ch k ( K m r / 2 +1 ,...,m r ) ≤ 244(1 − 1 5 r 1 / 3 ) r 2 ( k + log t 2 ) f ( r ) . Since r ≥ 4, we ha v e 244 /f ( r ) ≥ 244 and it follo ws from lemma 3.3 that (2) holds , as claimed. It is easy to c hec k that log 2 r Y j =3 (1 − 1 5 · 2 j / 3 ) ≥ 1 − log 2 r X j =3 1 5 · 2 j / 3 ≥ 1 − 1 10(1 − 2 − 1 / 3 ) , and therefore 244 /f ( r ) ≤ 474. If f o llo ws f rom (2) that f o r ev ery r which is a p o w er of 2 ch k ( K m 1 ,...,m r ) ≤ 474 r ( k + log t ) . (3) 11 Returning to the general case, assume that r ≥ 1. Cho ose an int eger r ′ whic h is a p o w er of 2 and r < r ′ ≤ 2 r . By applying (3), w e get ch k ( K m 1 ,...,m r ) ≤ ch k ( K m 1 ,...,m r , 2 , . . . , 2 | {z } r ′ − r ) ≤ 474 r ′ ( k + log m 1 + · · · + m r + 2( r ′ − r ) r ′ ) ≤ 948 r ( k + log m 1 + · · · + m r r ) , completing the pr oof. ✷ Denote K = K m, s, . . . , s | {z } r , where m ≥ 2 and s ≥ 2. Eve ry indu c ed subgraph of K has a v ertex of d eg ree at most r s , and th e refore by corollary 1.10 ch k ( K ) ≤ k ( rs + 1) for all k ≥ 1. Note that this upp er b ound for ch k ( K ) does not d epend of m , which means that a goo d lo w er b ound for ch k ( K m 1 ,...,m r ) has a more complicated form than the up per b ound giv en in th eorem 1.3. Pro of of Corollary 1.4 Let G = ( V , E ) b e a graph and k ≥ 1. De n ote r = χ ( G ), and let V = V 1 ∪ . . . ∪ V r b e a partition of the v ertices, suc h th at eac h V i is a stable set. Denote m i = | V i | for all i , 1 ≤ i ≤ r . By theorem 1.1 ch k ( G ) ≤ ch k ( K m 1 +1 ,...,m r +1 ) ≤ 948 r ( k + log m 1 + · · · + m r + r r ) = 948 χ ( G )( k + log ( | V | χ ( G ) + 1) ) , as needed. ✷ Pro of of C orollary 1.5 As p ro v ed by Bollob´ as in [6], for a fixed probabilit y p , 0 < p < 1, almost surely (i.e., with probabilit y that tends to 1 as n tends to infin it y), the r a n d om graph G n,p has c hromatic n umb er ( 1 2 + o (1)) log (1 / (1 − p )) n log n . By corollary 1.4, f o r every ǫ > 0 almost surely ch k ( G n,p ) ≤ 948( 1 2 + ǫ ) log (1 / (1 − p )) n log n ( k + log ( 3 log n log (1 / (1 − p )) + 1)) . The result follo ws sin ce k and p are constan ts. ✷ Note that in the pro o f of the last corollary w e ha ve not used a ny kno wledge concerning indep enden t sets of G n,p , as wa s done in [2 ] f o r the pro of of the sp ecial case. 12 4 Choice num b ers and orien tations Let D = ( V , E ) b e a d ig raph . W e denote the set of out-neigh b ors of v in D by N + D ( v ). A set of v ertices K ⊆ V is called a k ernel of D if K is an indep endent set and N + D ( v ) ∩ K 6 = ∅ for every v ertex v 6∈ K . Ric hard s o n’s th eorem (see, e.g., [5]) states that an y digraph w it h no o dd directed cycle has a k ernel. Pro of of T heo rem 1 .6 Let D = ( V , E ) be a digraph which con tains no o dd directed (simple) cycle and k ≥ 1. F or eac h v ∈ V , let S ( v ) b e a set of size k ( d + D ( v ) + 1). W e claim that the follo wing algorithm finds su bsets C ( v ) ⊆ S ( v ), wh ere | C ( v ) | = k f or all v ∈ V , and C ( u ) ∩ C ( v ) = ∅ for ev ery t w o adjacen t v ertices u, v ∈ V . 1. S ← ∪ v ∈ V S ( v ), W ← V and for ev ery v ∈ V , C ( v ) ← ∅ . 2. Cho ose a color c ∈ S ∩ ∪ v ∈ W S ( v ) and p u t S ← S − { c } . 3. Let K b e a kernel of the indu c ed sub g raph of D on the vertex set { v ∈ W : c ∈ S ( v ) } . 4. C ( v ) ← C ( v ) ∪ { c } for all v ∈ K . 5. W ← W − { v ∈ K : | C ( v ) | = k } . 6. If W = ∅ , stop. If not, go to step 2. During the algorithm, W is equal to { v ∈ V : | C ( v ) | < k } , and S is the set of remaining colors. W e first pro ve that in s t ep 2, S ∩ ∪ v ∈ W S ( v ) 6 = ∅ . When the algorithm reac hes step 2, it is ob vious that W 6 = ∅ . Sup p ose that w ∈ W in this step, and therefore | C ( w ) | < k . It follo ws easily from the definition of a ke rn el that every color fr o m S ( w ), wh ich has b een previously chosen in step 2, b elongs either to C ( w ) or to ∪ v ∈ N + D ( w ) C ( v ). Since | C ( w ) | + | [ v ∈ N + D ( w ) C ( v ) | < k + k · d + D ( v ) = | S ( w ) | , not all the colors of S ( w ) ha ve b een u sed. T h is means that S ∩ S ( w ) 6 = ∅ , as needed. It follo ws easily that the algorithm alw a ys terminates. Up on termination of the algorithm, | C ( v ) | = k f o r all v ∈ V . In step 4 the same color is assigned to the vertice s of a k ernel which is an indep endent set, and therefore C ( u ) ∩ C ( v ) = ∅ for every tw o adjacen t v ertices u, v ∈ V . This pro ves the correctness of the algorithm. 13 In step 4, the op eratio n C ( v ) ← C ( v ) ∪{ c } is p erformed for at least one v ertex. Upon termination | ∪ v ∈ V C ( v ) | ≤ k | V | , which means that the algorithm p erforms at most k | V | iterations. Th ere is a p olynomial time algorithm for finding a k ernel in a digraph with no o dd dir e cted cycle. Thus, the algorithm is of p olynomial time complexity in | V | and k , completing the pro of. ✷ Pro of of Corollary 1.7 T his is an immed iate consequence of theorem 1.6, since k ( d + D ( v ) + 1) ≤ k ( d + 1) for ev ery v ∈ V . ✷ Pro of of Corollary 1.8 The resu lt follo ws from 1.7 by taking the cyclic orient ation of the ev en cycle. ✷ The proof of corollary 1.9 is similar to the pro of of the sp ecial ca se which app ears in [8]. A graph G = ( V , E ) is k -de gr e e-cho os able if for ev ery f a mily of sets { S ( v ) : v ∈ V } , wh e re | S ( v ) | = k d ( v ) for all v ∈ V , there are subsets C ( v ) ⊆ S ( v ), where | C ( v ) | = k for all v ∈ V , and C ( u ) ∩ C ( v ) = ∅ for ev ery tw o adjacent v ertices u, v ∈ V . Lemma 4.1 If a gr aph G = ( V , E ) is c onne cte d, and G has a c onne cte d induc e d sub gr aph H = ( V ′ , E ′ ) which is k -de gr e e-cho osable, then G is k -de gr e e-cho osa ble. Pro of F or eac h v ∈ V , let S ( v ) b e a set of size k d ( v ). The pro of is by ind u ct ion on | V | . In case | V | = | V ′ | there is nothing to pr o v e. Assuming that | V | > | V ′ | , let v b e a ve rtex of G w hic h is at maximal distance from H . This guarant ees that G − v is co nn ec ted. Cho ose an y subset C ( v ) ⊆ S ( v ) suc h that | C ( v ) | = k , and remov e the colors of C ( v ) from all the ve rtices adjacent to v . The choic e can b e completed by applyin g the ind uctio n hyp othesis on G − v . ✷ Lemma 4.2 If c ≥ 2 , then Θ a,b,c is k -de gr e e-cho osable for e very k ≥ 1 . Pro of Supp ose that Θ a,b,c has v ertex set V = { u, v , x 1 , . . . , x a − 1 , y 1 , . . . , y b − 1 , z 1 , . . . , z c − 1 } and con tains the th r ee paths u − x 1 − · · · − x a − 1 − v , u − y 1 − · · · − y b − 1 − v , and u − z 1 − · · · − z c − 1 − v . F or eac h w ∈ V , let S ( w ) b e a set of size k d ( w ). F or the vertex u we c ho ose a su b set C ( u ) ⊆ S ( u ) − S ( z 1 ) of size k . F or eac h no de according to the sequence x 1 , . . . , x a − 1 , y 1 , . . . , y b − 1 , v , z c − 1 , . . . , z 1 w e choose a subs e t of k colors that w ere n ot c hosen in adjacen t earlier no des. ✷ F or the p r oof of corollary 1.9, we shall need the f ollo win g lemma which app ears in [8]. Lemma 4.3 If ther e is no no de which disc onne cts G , then G is an o dd cycle, or G = K n , or G c onta ins, as a no de induc e d sub gr aph, an ev en cycle without chor d or with only one c hor d. 14 Pro of of Corollary 1.9 Supp ose that a connected graph G is not K n , and n ot an o dd cycle. If G is n o t a regular graph, then ev ery induced subgraph of G has a ve rtex of d eg ree at most ∆( G ) − 1, and by corollary 1.12 ch k ( G ) ≤ k ∆( G ) for all k ≥ 1. If G is a regular graph, then there is a part of G not disconnected by a no de, whic h is n ei ther an o dd cycle nor a complete graph. It follo ws from lemma 4.3 that G con tains, as a no de induced subgraph , an eve n cycle or a p a rticular kind of Θ a,b,c graph. W e know f rom corollary 1.8 and lemma 4.2 that b oth an ev en cycle and Θ a,b,c are k -degree -choosable for every k ≥ 1. Th e r e su lt follo ws from lemma 4.1. ✷ Pro of of Corollary 1.10 It is p r o v ed in [4 ] th a t a graph G = ( V , E ) has an orientati on D in whic h ev ery outdegree is at most d if and only if M ( G ) ≤ d . Therefore, there is an orienta tion D of G in whic h the m aximum outdegree is at most ⌈ M ( G ) ⌉ . S ince D cont ains n o o dd directed cycles, the result follo ws from corollary 1.7. ✷ Pro of of Corollary 1.11 M ( G ) ≤ 2, since any bipartite (simple) graph on r vertic es con tains at most 2 r − 2 edges. The resu lt follo ws from corollary 1.10. ✷ Pro of of Corollary 1.12 W e claim that if every ind uced subgraph of a graph G = ( V , E ) h as a v ertex of degree at m o st d , then G h a s an acyclic orienta tion in which the maxim um outdegree is d . The pro of is by indu ct ion on | V | . If | V | = 1, the result is trivial. If | V | > 1, let v b e a verte x of G with degree at most d . By th e induction hyp ot hesis, G − v has an acyclic orien tation in whic h th e maxim um outdegree is d . W e complete this orienta tion of G − v by orienting ev ery edge incident to v fr om v to its appropr ia te neigh b or and obtain the desired orien tation of G , as claimed. The result follo ws from corollary 1.7. ✷ An undirected graph G is called triangulate d if G do es not con tain an induced subgraph iso- morphic to C n for n ≥ 4. Being triangulated is a h er ed it ary p r operty inherited by all the indu ce d subgraphs of G . A ve rtex v of G is called simplicial if its adjacency set Ad j ( v ) ind uces a complete subgraph of G . It is pro ved in [12] that every triangulated graph has a simplicial vertex. Pro of of Corollary 1.13 Su pp o se th a t G is a triangulated graph, and let H b e an ind uced subgraph of G . Since H is triangulated, it has a simplicia l v ertex v . Th e set of vertice s { v } ∪ Ad j H ( v ) induces a complete su bgraph of H , and t h er efore v has degree at most ω ( G ) − 1 in H . It follo ws from corollary 1.10 that ch k ( G ) ≤ k ω ( G ) f or ev ery k ≥ 1. F or ev ery graph G and k ≥ 1, ch k ( G ) ≥ k ω ( G ) and hence ch k ( G ) = k ω ( G ) for ev ery k ≥ 1. Since G is triangulated, it is also p erfect, wh ic h means that χ ( G ) = ω ( G ), as needed. ✷ 15 Pro of of Corollary 1.14 It is easy to see that L ( G ) is triangulated if and only if G con tains no C n for ev ery n ≥ 4. The result follo ws f rom corollary 1.13. ✷ The v al idity of the list-c hr oma tic conjecture for graphs of class 2 with m a ximum degree 3 (and in particular for snarks) follo ws easily fr o m corollary 1.9. Supp ose that G is a graph of class 2 with ∆( G ) = 3. Let C b e a connected comp onen t of L ( G ). If C is not a complete graph, and not an o dd cycle, then ch ( C ) ≤ ∆( C ) ≤ ∆ ( L ( G )) ≤ 4. If C is a complete graph or an o dd cycle, then it is easy to see that ∆( C ) ≤ 2, and therefore by corollary 1.10 ch ( C ) ≤ ∆( C ) + 1 ≤ 3. It follo ws that ch ( L ( G )) ≤ 4. Since G is a graph of class 2, ch ( L ( G )) ≥ χ ( L ( G )) = ∆( G ) + 1 = 4, and h ence , ch ( L ( G )) = χ ( L ( G )) = 4. 5 Prop erties of (2 k : k ) -c ho osable graphs Let A and B b e sets of size 4. W e den o te p ( A, B ) = { ( C , D ) : C ⊆ A, D ⊆ B , | C | = | D | = 2 } . Supp ose that S ⊆ p ( A 1 , B 1 ) and that T ⊆ p ( A 2 , B 2 ). W e sa y that S and T are isomorph ic if th e re exist t w o bijections f : A 1 7→ A 2 and g : B 1 7→ B 2 so that ( C , D ) ∈ S iff ( f ( C ) , g ( D )) ∈ T for every C ⊆ A and D ⊆ B , where | C | = | D | = 2. Let A and B b e sets of size 4, and supp ose that S ⊆ p ( A, B ). Supp ose that H 1 , . . . , H 6 are all th e sub s e ts of A of siz e 2. F or eac h i , 1 ≤ i ≤ 6, w e denote c ( H i ) = { G : ( H i , G ) ∈ S } and d i = | c ( H i ) | . The s e qu en c e ( d 1 , . . . , d 6 ) is called the degree sequence of S . W e sa y that S is s pecial if it has the follo wing prop erties: 1. Its d egree sequence is (6 , 5 , 5 , 3 , 3 , 1). 2. If H and G are the tw o subs e ts of A for which | c ( H ) | = | c ( G ) | = 3, then | H ∩ G | = 1. Denote H = { 1 , 2 } , G = { 1 , 3 } , an d A = { 1 , 2 , 3 , 4 } . 3. c ( H ) = c ( G ). 4. c ( H ) has either th e form {{ 5 , 6 } , { 5 , 7 } , { 5 , 8 }} or the form {{ 5 , 6 } , { 5 , 7 } , { 6 , 7 }} . 5. Either | c ( { 2 , 3 } ) | = 1 and | c ( { 1 , 4 } ) | = 6, or | c ( { 2 , 3 } ) | = 6 and | c ( { 1 , 4 } ) | = 1. W e sa y that S has p roperty P 1 iff comp ( H ) has the form {{ 5 , 6 } , { 5 , 7 } , { 5 , 8 }} and that it has prop ert y P 2 iff | comp ( { 2 , 3 } ) | = 1. 16 Supp ose that K 2 , 2 has v ertex set V = X ∪ Y , where X = { x 1 , x 2 } , Y = { y 1 , y 2 } , and it has exactly the edges { x i , y j } . F or eac h v ∈ V , let S ( v ) b e a set of size 4. By C ( v ) we denote a subset of S ( v ) of size 2. W e sa y that C ( x 1 ) and C ( x 2 ) are compatible if there exist tw o sub s e ts C ( y 1 ) and C ( y 2 ), so that C ( u ) ∩ C ( v ) = ∅ for ev ery t w o adjacent vertice s u, v ∈ V . A subset C ( x 1 ) ⊆ S ( x 1 ) is called bad if C ( x 1 ) is not compatible w it h any C ( x 2 ). An analogous definition is m a de for C ( x 2 ). W e sa y th a t a family of sets { S ( v ) : v ∈ V } is d e fected if there exist t wo bad subsets C ( x 1 ) and C ( x 2 ). W e denote by incomp ( x 1 , x 2 ) the s e t of incompatible pairs ( C ( x 1 ) , C ( x 2 )). Lemma 5.1 If the family of sets { S ( v ) : v ∈ V } is defe cte d and C ( x 1 ) is b ad, then b oth S ( y 1 ) and S ( y 2 ) interse ct C ( x 1 ) and at le ast one them c ontains C ( x 1 ) . Pro of Su pp ose that neither S ( y 1 ) nor S ( y 2 ) co ntain C ( x 1 ). R emov e the co lors of C ( x 1 ) from S ( y 1 ) and S ( y 2 ). No w b oth S ( y 1 ) and S ( y 2 ) h av e size at least 3. W e can assu me the worst case, in wh ich b oth S ( y 1 ) and S ( y 2 ) are sub sets of S ( x 2 ), and th e refore | S ( y 1 ) ∩ S ( y 2 ) | ≥ 2. Let C b e a subs et of S ( y 1 ) ∩ S ( y 2 ) of size 2. C h oose a sub s e t C ( x 2 ) ⊆ S ( x 2 ) − C . W e hav e that C ( x 1 ) and C ( x 2 ) are compatible in con trast to the fact that C ( x 1 ) is bad . This prov es that at least one of S ( y 1 ) and S ( y 2 ) con tains C ( x 1 ). Supp ose that S ( y 1 ) ∩ C ( x 1 ) = ∅ . Cho ose a su b set C ( y 2 ) ⊆ S ( y 2 ) − C ( x 1 ) and a sub s e t C ( x 2 ) ⊆ S ( x 2 ) − C ( y 2 ). W e ha ve that C ( x 1 ) and C ( x 2 ) are compatible in con trast to the fact that C ( x 1 ) is bad. This prov es that b oth S ( y 1 ) and S ( y 2 ) inte rsect C ( x 1 ). ✷ Lemma 5.2 If the family of sets { S ( v ) : v ∈ V } is defe cte d, then b oth S ( x 1 ) and S ( x 2 ) c ontain exactly one b ad subset. F urthermor e, at le ast one of the fol lowing is valid: 1. The set incomp ( x 1 , x 2 ) is sp e cial and has pr op erties P 1 and P 2 . 2. incomp ( x 1 , x 2 ) has de gr e e se quenc e (6 , 5 , 5 , 3 , 2 , 2) . 3. | incomp ( x 1 , x 2 ) | = 21 . Pro of The set S ( x 1 ) cont ains a bad subset, wh ich we denote b y C ( x 1 ) = { 1 , 2 } . Without loss of generalit y , w e can assume b y lemma 5.1 that C ( x 1 ) ⊆ S ( y 1 ) and that S ( y 2 ) in tersects C ( x 1 ). Denote S ( y 1 ) = { 1 , 2 , 3 , 4 } . Sin ce C ( x 1 ) is bad, we m ust h av e that | ( S ( y 1 ) ∩ S ( y 2 )) − C ( x 1 ) | < 2. Case 1: C ( x 1 ) ⊆ S ( y 2 ) and | S ( y 1 ) ∩ S ( y 2 ) | = 3. 17 Denote S ( y 2 ) = { 1 , 2 , 3 , 5 } . Sin ce C ( x 1 ) is b ad, surely { 3 , 4 , 5 } ⊆ S ( x 2 ). T he s et S ( x 2 ) conta ins a bad sub s e t, whic h we denote by C ( x 2 ), and therefore { 1 , 2 } ∩ S ( x 2 ) 6 = ∅ . W e can assume, without loss of generalit y , that S ( x 2 ) = { 1 , 3 , 4 , 5 } . Since C ( x 2 ) is bad and | S ( y 1 ) ∩ S ( y 2 ) | = 3, we m u st ha v e that C ( x 2 ) ⊆ S ( y 1 ) ∩ S ( y 2 ). Hence, C ( x 2 ) = { 1 , 3 } and S ( x 1 ) = { 1 , 2 , 4 , 5 } . W e hav e that S ( x 1 ) = { 1 , 2 , 4 , 5 } , S ( x 2 ) = { 1 , 3 , 4 , 5 } , S ( y 1 ) = { 1 , 2 , 3 , 4 } , S ( y 2 ) = { 1 , 2 , 3 , 5 } . The set incomp ( x 1 , x 2 ) is sp ecial and has p roperties P 1 and P 2 . Case 2: C ( x 1 ) ⊆ S ( y 2 ) and | S ( y 1 ) ∩ S ( y 2 ) | = 2. Denote S ( y 2 ) = { 1 , 2 , 5 , 6 } . Since C ( x 1 ) is bad, surely | S ( x 2 ) ∩ { 3 , 4 , 5 , 6 }| ≥ 3. Supp ose without loss of generalit y that { 3 , 4 , 5 } ⊆ S ( x 2 ). The set S ( x 2 ) conta ins a b ad subset, whic h w e d enot e b y C ( x 2 ), and therefore { 1 , 2 } ∩ S ( x 2 ) 6 = ∅ . W e can assume, without loss of generalit y , th a t S ( x 2 ) = { 1 , 3 , 4 , 5 } . Since C ( x 2 ) is b a d and | S ( y 1 ) ∩ S ( y 2 ) | = 2, we m us t h av e th a t C ( x 2 ) ∩ { 1 , 2 } 6 = ∅ , and therefore 1 ∈ C ( x 2 ). W e can assum e , w ith ou t loss of generalit y , that C ( x 2 ) = { 1 , 3 } . Since C ( x 2 ) is bad, w e m ust hav e that 4 ∈ S ( x 1 ) and S ( x 1 ) ∩ { 5 , 6 } 6 = ∅ . Supp ose without loss of ge n er ality that S ( x 1 ) = { 1 , 2 , 4 , 5 } . Th is is a con tradiction to the fact that C ( x 2 ) is bad. Case 3: | C ( x 1 ) ∩ S ( y 2 ) | = 1 and | S ( y 1 ) ∩ S ( y 2 ) | = 2. W e can assume, without loss of generalit y , th a t 1 ∈ S ( y 2 ). Denote S ( y 2 ) = { 1 , 3 , 5 , 6 } . Since C ( x 1 ) is bad, su rely S ( x 2 ) = { 3 , 4 , 5 , 6 } . The set S ( x 2 ) con tains a bad subset, which we d e note b y C ( x 2 ). Since C ( x 2 ) is b a d and | S ( y 1 ) ∩ S ( y 2 ) | = 2, we m us t h av e th a t C ( x 2 ) ∩ { 1 , 3 } 6 = ∅ , and therefore 3 ∈ C ( x 2 ). If C ( x 2 ) = { 3 , 4 } , then we must h a v e that S ( x 1 ) = { 1 , 2 , 5 , 6 } , s o S ( x 1 ) = { 1 , 2 , 5 , 6 } , S ( x 2 ) = { 3 , 4 , 5 , 6 } , S ( y 1 ) = { 1 , 2 , 3 , 4 } , S ( y 2 ) = { 1 , 3 , 5 , 6 } . The set incomp ( x 1 , x 2 ) has degree sequence (6 , 5 , 5 , 3 , 2 , 2). Otherwise, sup pose without loss of generalit y that C ( x 2 ) = { 3 , 5 } . W e m us t ha ve that S ( x 1 ) = { 1 , 2 , 4 , 6 } , s o S ( x 1 ) = { 1 , 2 , 4 , 6 } , S ( x 2 ) = { 3 , 4 , 5 , 6 } , S ( y 1 ) = { 1 , 2 , 3 , 4 } , S ( y 2 ) = { 1 , 3 , 5 , 6 } . In this case | incomp ( x 1 , x 2 ) | = 21. Case 4: | C ( x 1 ) ∩ S ( y 2 ) | = 1 and | S ( y 1 ) ∩ S ( y 2 ) | = 1. W e can assume, without loss of generalit y , th a t 1 ∈ S ( y 2 ). Denote S ( y 2 ) = { 1 , 5 , 6 , 7 } . Since C ( x 1 ) is bad, we must ha ve that S ( x 2 ) ∩ { 3 , 4 } 6 = ∅ and | S ( x 2 ) ∩ { 5 , 6 , 7 }| ≥ 2. Supp ose w it hout loss of 18 generalit y that { 3 , 5 , 6 } ⊆ S ( x 2 ). Since C ( x 1 ) is bad, we m ust ha ve that either S ( x 2 ) = { 4 , 3 , 5 , 6 } or S ( x 2 ) = { 7 , 3 , 5 , 6 } . It is easy to s e e that in b oth cases we ha ve a con tradiction to the fact that S ( x 2 ) con tains a bad subs e t. ✷ F or eve ry i , 1 ≤ i ≤ m , let A i b e a sequ e n ce of 4 d istin ct elemen ts. T he sequence A 1 , . . . , A m is called v alid if wh e never c ∈ A i ∩ A i +1 , then c app ears in th e same p osition in b oth A i and A i +1 . A v alid sequ e nce A 1 , . . . , A m is called legal if whenev er c ∈ A i +1 − A i , then c 6∈ A j for ev ery j , 1 ≤ j ≤ i . By a subsequ e n ce of A 1 , . . . , A m w e mean a sequence of th e form A i , A i +1 , . . . , A j , where 1 ≤ i ≤ j ≤ m . Let A 1 , . . . , A m b e a v alid sequ e nce. Th e p air ( A i , A i +1 ) con tains a c hange in the k th p osition if the elemen ts wh ic h app ear in the k th p osition of A i and A i +1 are different. The sequ e n ce A 1 , . . . , A m con tains a c hange in the k th p osition if there exists a pair ( A i , A i +1 ) which cont ains a c hange in the k th p osition. Let A 1 , . . . , A m b e a sequ e n ce. By C i w e denote a su bset of A i of size 2. W e s a y that C 1 and C m are compatible if there exist subsets { C k : 1 < k < m } so that C p ∩ C p +1 = ∅ for ev ery p , 1 ≤ p < m . A s ubset C 1 is called bad if C 1 is not compatible with any C m . A su bset C 1 is called go o d if C 1 is compatible with ev ery C m . W e denote by comp ( C 1 ; A 1 , . . . , A m ) the set whic h consists of all the sub set s C m whic h are compatible with C 1 , and by comp ( A 1 , . . . , A m ) the set of all the compatible pairs ( C 1 , C m ). By g ood ( A 1 , . . . , A m ) we denote the set w hic h consists of all the go od subsets that A 1 con tains. Lemma 5.3 If the valid se quenc e D 1 , . . . , D r c onta ins a change in at le ast 3 p ositions and ther e is no i , 1 < i < r − 1 , for which D i = D i +1 , then it c ontains a subse quenc e A 1 , . . . , A m , so that the se quenc e A 1 c onta ins at le ast one go o d subset. F urthermor e, the se quenc e A 1 , . . . , A m has at le ast one of the f ol lowing pr op erties: 1. | g ood ( A 1 , . . . , A m ) | ≥ 3 . 2. | comp ( A 1 , . . . , A m ) | > 23 . 3. The set comp ( A 1 , . . . , A m ) i s sp e cial. If m is o dd , then comp ( A 1 , . . . , A m ) has exactly one of the pr op erties P 1 and P 2 . If m is even then comp ( A 1 , . . . , A m ) has either b oth or none of the pr op erties P 1 and P 2 . 19 Pro of W e consider the follo wing cases. Case 1: F or some i , | D i ∩ D i +1 | ≤ 1. In this case | g ood ( D i , D i +1 ) | ≥ 3. Case 2: F or every j , | D j ∩ D j +1 | ≤ 2, and for some i , | D i ∩ D i +1 | = 2. Assume without loss of generalit y that the pair ( D i , D i +1 ) con tains a c hange in th e first and second p ositions. A t least one of the pairs ( D i − 1 , D i ) and ( D i , D i +1 ) conta ins a change in some p osition. Supp ose that th e pair ( D i − 1 , D i ) con tains a c hange in some p osition. The p roof in case the pair ( D i , D i +1 ) con tains a c hange in some p osition is s im ilar. If the pair ( D i − 1 , D i ) conta ins a change in at least one of the firs t and s econd p ositions, then s urely | g ood ( D i − 1 , D i , D i +1 ) | ≥ 3. If the only p osition in w hic h the pair ( D i − 1 , D i ) conta ins a change is either the third or the fourth p osition, then comp ( D i − 1 , D i , D i +1 ) is sp ecial, h as p roperty P 2 , an d do es n ot ha ve prop ert y P 1 . If the p a ir ( D i − 1 , D i ) con tains a c hange in the thir d and fourth p ositions, then | comp ( D i − 1 , D i , D i +1 ) | = 27. Case 3: F or every j , | D j ∩ D j +1 | ≤ 1. Let B 1 , . . . B k b e a subsequen c e of D 1 , . . . , D r whic h con tains a change in at least 3 p ositions, b ut no prop er s ubsequence of B 1 , . . . , B k has this prop ert y . This implies that the three pairs ( B 1 , B 2 ), ( B 2 , B 3 ) a nd ( B k − 1 , B k ) c ontai n a c hange in three differen t p ositi ons. W e can assume, without loss of ge ner ality , th at the three pairs con tain a c hange in the first, second and third p osit ions resp ectiv ely . Su pp o se that 2 ≤ i ≤ k − 2, and consider th e pair ( B i , B i +1 ). If th is pair con tains a c hange in the first p osition, then the sequence B 2 , . . . , B m con tains a c hange in at least 3 p ositions. If this pair cont ains a c hange in the third or fourth p osition, then th e sequence B 1 , . . . , B i +1 con tains a c hange in at lea st 3 p ositi ons. Hence, the pair ( B i , B i +1 ) con tains a c hange in the second p o sition. If k = 4 then the set c omp ( B 1 , . . . , B 4 ) is sp ecial and do es not h a v e neit h er prop ert y P 1 nor prop ert y P 2 . If k > 4 then | g ood ( B 1 , . . . , B 4 ) | = 3. ✷ Lemma 5.4 If the set comp ( A 1 , . . . , A m ) is sp e cial, then b oth the set comp ( A 1 , A 1 , . . . , A m ) and the se t comp ( A 1 , . . . , A m , A m ) a r e sp e cial. The set comp ( A 1 , A 1 , . . . , A m ) h as pr op erty P 1 iff the set comp ( A 1 , . . . , A m ) has pr op erty P 1 . The set comp ( A 1 , A 1 , . . . , A m ) has pr op erty P 2 iff the set comp ( A 1 , . . . , A m ) do es not have pr op erty P 2 . The set comp ( A 1 , . . . , A m , A m ) has pr op erty P 1 iff the set comp ( A 1 , . . . , A m ) do es not have pr op erty P 1 . The set comp ( A 1 , . . . , A m , A m ) has pr op erty P 2 iff the set comp ( A 1 , . . . , A m ) has pr op erty P 2 . 20 Lemma 5.5 If A 1 , A 2 , A 2 , A 3 is a le gal se quenc e, then comp ( A 1 , A 2 , A 2 , A 3 ) = comp ( A 1 , A 3 ) . Pro of Let k 1 , . . . k n b e all the p ositions in whic h A 1 , A 2 , A 2 , A 3 do es not con tain a change. It is easy to v erify that ( C , D ) ∈ comp ( A 1 , A 3 ) iff there is no i for w h ic h C conta ins the k i th element of A 1 and D con tains the k i elemen t of A 3 . The s a me prop ert y holds also for comp ( A 1 , A 2 , A 2 , A 3 ). ✷ Lemma 5.6 If A i , . . . , A j is a subse quenc e of A 1 , . . . , A m , then | comp ( A 1 , . . . , A m ) | ≥ | comp ( A i , . . . , A j ) | . Pro of By in duction on m . If m = j − i + 1, there is nothing to p ro v e. Supp ose that m > j − i + 1. Assume that i > 1. The p roof in case j < m is similar. Hence, | comp ( A 1 , . . . , A m ) | ≥ | comp ( A 2 , A 2 , . . . , A m ) | = | comp ( A 2 , . . . , A m ) | ≥ | comp ( A i , . . . , A j ) | , where the last inequalit y follo ws f rom the ind u ct ion hyp othesis. ✷ Lemma 5.7 If A i , . . . , A j is a subse quenc e of A 1 , . . . , A m , then | g ood ( A 1 , . . . , A m ) | ≥ | g ood ( A i , . . . , A j ) | . Pro of Similar to the pro of of lemma 5.6. ✷ Lemma 5.8 Supp ose that i ≥ 0 , and denote by F the se quenc e A i +1 , . . . , A m to ge th er with an ad- ditional A i +1 as the first element of the se quenc e in c ase i ≡ 1 (mo d 2) . If A i +1 , . . . , A m is a sub- se quenc e of A 1 , . . . , A m and | comp ( A i +1 , . . . , A m ) | = | comp ( A 1 , . . . , A m ) | , then comp ( A 1 . . . , A m ) is isomorphic to comp ( F ) . Pro of W e can assume that A 1 , . . . , A m is a v alid sequence. Sup pose that i ≡ 1 (mo d 2). The pro of in case i ≡ 0 (mo d 2) is similar. Supp ose that C 1 ⊆ A 1 . Denote by T the sub set of A i +1 that app ears in the tw o p ositions in whic h C 1 do es not app ear in A 1 . Since A 1 , . . . , A i +1 is a v alid sequence, we h av e that C 1 is compatible with T . Hence, V = comp ( C 1 ; A 1 , . . . , A m ) ⊇ comp ( T ; A i +1 , . . . , A m ) = W . Since | comp ( A i +1 , . . . , A m ) | = | comp ( A 1 , . . . , A m ) | , we must ha ve that V = W . It is easy to see no w that comp ( A 1 , . . . , A m ) is isomorphic to comp ( A i +1 , A i +1 , . . . , A m ). ✷ 21 Lemma 5.9 Supp ose that i, j ≥ 0 . Denote b y F the se qu enc e A i +1 , . . . , A m − j to ge th er with an additiona l A i +1 as the first element of the se quenc e in c ase i ≡ 1 (mo d 2) and an additional A m − j as the last element of the se quenc e in c ase j ≡ 1 (mo d 2) . If A i +1 , . . . , A m − j is a subse quenc e of A 1 , . . . , A m and | comp ( A i +1 , . . . , A m − j ) | = | comp ( A 1 , . . . , A m ) | , then comp ( A i +1 . . . , A m − j ) is isomorph ic to comp ( F ) . Pro of Apply lemma 5.8 t wice. ✷ Lemma 5.10 Supp ose that r is o dd and that r ≥ 3 . If the v alid se quenc e D 1 , . . . , D r c onta ins a change in at le ast 3 p ositions, then the se quenc e D 1 c onta ins at le ast one go o d subset. F urthermor e, at le ast one of the fol lowing is v a lid: 1. | g ood ( D 1 , . . . , D r ) | ≥ 3 . 2. | comp ( D 1 , . . . , D r ) | > 23 . 3. The set comp ( D 1 , . . . , D m ) is sp e cial and has exactly one of the pr op erties P 1 and P 2 . Pro of W e can assume, without loss of generalit y , that D 1 , . . . , D r is legal. Due to lemma 5.5, we can assume th a t there is n o i , 1 < i < r − 1, for whic h D i = D i +1 . It follo w s from lemma 5.3 that the s equ e n ce D 1 , . . . , D r con tains a sub sequence A 1 , . . . , A m , so th a t the sequence A 1 con tains at least o ne goo d subs et. It follo ws from lemma 5.7 that | g ood ( D 1 , . . . , D r ) | ≥ 1. According to lemma 5.3, we consider the follo wing cases: Case 1: | g ood ( A 1 , . . . , A m ) | ≥ 3. It follo ws from lemma 5.7 th a t | g ood ( D 1 , . . . , D r ) | ≥ 3. Case 2: | comp ( A 1 , . . . , A m ) | ≥ 27. It follo ws from lemma 5.6 th a t | comp ( D 1 , . . . , D r ) | > 23. Case 3: Th e set comp ( A 1 , . . . , A m ) is sp ecial. W e kn ow th at if m is o dd, then comp ( A 1 , . . . , A m ) has exactly one of the pr o p erties P 1 and P 2 . F ur thermore, if m is ev en then comp ( A 1 , . . . , A m ) has either b oth or none of the prop erties P 1 and P 2 . If | comp ( D 1 , . . . , D r ) | > | comp ( A 1 , . . . , A m ) | , then | comp ( D 1 , . . . , D m ) | > 23. Supp ose that | comp ( D 1 , . . . , D r ) | = | comp ( A 1 , . . . , A m ) | . I t follo ws f rom lemma 5.9 that comp ( D 1 , . . . , D r ) is isomorphic to comp ( F ) for some sequence F . Since r is o dd and using lemma 5.4, it is easy to see that comp ( D 1 , . . . , D r ) is sp ecial and has exactly one of the prop erties P 1 and P 2 . ✷ 22 Pro of of Theorem 1.16 It is easy to see th a t a graph G is (4 : 2)-c hoosable iff its core is (4 : 2)- c ho osable. Du e to theorem 1.15, we n ee d to p ro v e that for ev ery m ≥ 1, Θ 2 , 2 , 2 m is (4 : 2)-c hoosable. Supp ose that m is o dd and that m ≥ 3. Assume that Θ 2 , 2 ,m − 1 has v ertex set V = { u, v , z 1 , . . . , z m } and conta ins the three paths z 1 − z 2 − · · · − z m , z 1 − u − z m , and z 1 − v − z m . F or eac h w ∈ V , let S ( w ) b e a set of s ize 4. W e denote A i = S ( z i ) for eve ry i , 1 ≤ i ≤ m . W e can assume that A 1 , . . . , A m is a v alid sequence. Supp ose first that the sequence A 1 , . . . , A m con tains a change in at most 2 p ositions. This means th a t there is a set C of size 2 so th a t C ⊆ A i for ev ery i , 1 ≤ i ≤ m . F rom A i when i is o dd, c ho ose the subset C . Complete the choic e b y c ho osing a sub set of S ( w ) − C for ev ery other v ertex w . Supp ose next t h at the sequence A 1 , . . . , A m con tains a c hange in at least 3 p ositions. T he graph induced b y the set of vertices { z 1 , z m , u, v } = W is isomorp hic to K 2 , 2 . Denote x 1 = z 1 , x 2 = z m , y 1 = u , and y 2 = v . W e u s e the same terminology as b efore. Case 1: { S ( w ) : w ∈ W } is not defected. Supp ose with ou t loss of generalit y that S ( z 1 ) con tains no bad su bset. If follo ws f rom lemma 5.10 that | g ood ( A 1 , . . . , A m ) | ≥ 1, and th e refore a c hoice is p ossible. Case 2: { S ( w ) : w ∈ W } is defected. According to lemma 5.10, w e consider the follo wing cases: Case 2a: | g ood ( A 1 , . . . , A m ) | ≥ 3. It follo ws from le mma 5.2 that S ( z 1 ) cont ains exactly one bad sub set, and therefore a choice is p ossible. Case 2b: | comp ( D 1 , . . . , D r ) | > 23. It follo ws from lemma 5.2 th a t | incomp ( z 1 , z m ) | ≤ 23, and therefore a c hoice is p ossible. Case 2c: T he set comp ( D 1 , . . . , D m ) is sp ecial. W e kn o w that comp ( D 1 , . . . , D m ) has exactly one of the prop erties P 1 and P 2 . It is easy to see from lemma 5.2 that th e set incomp ( z 1 , z m ) do es not cont ain the set comp ( D 1 , . . . , D m ), and therefore a c hoice is p ossible. ✷ Pro of of Theorem 1.17 Sup pose that G = ( V , E ) is (2 mk : mk )-c ho o sable for k o dd. W e pr o v e that G is 2 m -c ho osable as well . F or eac h v ∈ V , le t S ( v ) b e a s e t of size 2 m . With ev ery color c w e associate a set F ( c ) of size k , s u c h th at F ( c ) ∩ F ( d ) = ∅ if c 6 = d . F or ev ery v ∈ V , w e 23 define T ( v ) = ∪ c ∈ S ( v ) F ( c ). Since G is (2 mk : mk )- choosable, there are subsets C ( v ) ⊆ T ( v ), wher e | C ( v ) | = mk for all v ∈ V , and C ( u ) ∩ C ( v ) = ∅ for every t w o adjacen t v ertices u, v ∈ V . Fix a v ertex v ∈ V . S ince k is o dd, there is a color c ∈ S ( v ) for wh ic h | C ( v ) ∩ F ( c ) | > k / 2, so we d e fin e f ( v ) = c . In case u and v are adjacen t v ertices for whic h c ∈ S ( u ) ∩ S ( v ), it is not p ossible that b oth | C ( u ) ∩ F ( c ) | and | C ( v ) ∩ F ( c ) | are greater than k / 2. This pro ves that f is a prop er vertex-c oloring of G assigning to eac h vertex v ∈ V a color in S ( v ). ✷ 6 The comp l exit y of graph c ho osabilit y Let G = ( V , E ) b e a graph. W e den o te by G ′ the graph obtained from G by addin g a n e w vertex to G , and j oinin g it to every v ertex in V . Consider the follo wing d ecision problem: GRAPH k -COLORABILIT Y INST ANCE: A graph G = ( V , E ). QUESTION: Is G k -colorable? The standard tec hn ique to sh ow a p olynomial transformation f r om GRAPH k -COLORABILITY to GRAPH ( k + 1)-COLORABILITY is to us e the fact that χ ( G ′ ) = χ ( G ) + 1 for ev ery graph G . Ho w ev er, it is not true that ch ( G ′ ) = ch ( G ) + 1 for ev ery graph G . T o s ee that, w e first pr o v e that K ′ 2 , 4 is 3-c ho osable. Supp ose that K ′ 2 , 4 has v ertex set V = { v , x 1 , x 2 , y 1 , y 2 , y 3 , y 4 } , and conta ins exactly th e ed g es { x i , y j } , { v , x i } , and { v , y j } . F or eac h w ∈ V , let S ( w ) b e a set of size 3. Case 1: All th e sets are the same. A c hoice can b e made since K ′ 2 , 4 is 3-colorable. Case 2: Th e re is a set S ( x i ) whic h is not equal to S ( v ). Without loss of generalit y , sup p ose th a t S ( v ) 6 = S ( x 1 ). F or the n ode v , choose a color c ∈ S ( v ) − S ( x 1 ), and remov e c from the sets of the other vertic es. W e can assum e that every set S ( y j ) is of size 2 now. Supp ose first that S ( x 1 ) and S ( x 2 ) are d isjoi nt. The num b er of differen t sets consisting of one color from eac h of th e S ( x i ) is at least 6, and therefore we can choose colors c i ∈ S ( x i ), su c h th a t { c 1 , c 2 } do es not app ear as a set of S ( y j ). W e complete the c hoice by choosing for eve ry ve rtex y j 24 a color f rom S ( y j ) − { c 1 , c 2 } . Supp ose n ext that c ∈ S ( x 1 ) ∩ S ( x 2 ). F or eve ry ve rtex x i w e choose c , and for every v ertex y j w e choose a color from S ( y j ) − { c } . Case 3: Th e re is a set S ( y j ) whic h is not equal to S ( v ). Without loss of generalit y , supp ose th at S ( v ) 6 = S ( y 1 ). F or the no de v , c ho ose a color c ∈ S ( v ) − S ( y 1 ), an d remov e c from the sets of the other ve rtices. Supp ose fir st that S ( x 1 ) and S ( x 2 ) are disjoin t. Th e num b er of different sets consisting of one color from eac h of the S ( x i ) is at least 4, and since | S ( y 1 ) | = 3 we can choose colo rs c i ∈ S ( x i ), suc h that S ( y j ) − { c 1 , c 2 } 6 = ∅ for ev ery v ertex y j . W e can complete the choi ce as in case 2. In case S ( x 1 ) and S ( x 2 ) are not disjoin t, we pro ceed as in case 2. This completes th e proof that K ′ 2 , 4 is 3-c ho osable. I t follo ws f rom th eorem 1.15 and corol- lary 1.12 that ch ( K 2 , 4 ) = 3, and therefore ch ( K ′ 2 , 4 ) = ch ( K 2 , 4 ) = 3. The f o llo win g lemma exhibits a construction w hic h in crea ses the c hoice num b er of a graph in exactly 1. Lemma 6.1 L et G = ( V , E ) b e a gr aph. If H is the disjoint uni on of | V | c opies of G , then ch ( H ′ ) = ch ( G ) + 1 . Pro of Let H b e the disjoint u n io n of the graphs { G i : 1 ≤ i ≤ | V |} , wh e re eac h G i is a cop y of G . Supp ose that H ′ is obtained from H b y joining the new vertex v to all the vertic es of H . W e claim that if G is k -c ho osable, then H ′ is ( k + 1)-c ho osable. F or eac h w ∈ V ( H ′ ), let S ( w ) b e a set of size k + 1. Ch oose a color c ∈ S ( v ), and remo ve c from th e s e ts of the other vertice s. W e can complete th e choic e since G is k -c ho osa ble. W e n o w pro ve that if H ′ is k -c ho osa b le, then G is ( k − 1)-c ho osable. It is easy to see that this is true w hen G is a complete graph. If G is not a complete graph, then by corollary 1.9 ch ( G ) < | V | , and therefore ch ( H ′ ) ≤ | V | . Hence, w e can assu m e that k ≤ | V | . F or eac h w ∈ V , let S ( w ) b e a set of s iz e k − 1, s uc h that S ( w ) ∩ { 1 , 2 , . . . , | V |} = ∅ . F or every i , 1 ≤ i ≤ | V | , on the v ertices of the graph G i w e put the sets S ( w ) together with the add iti onal color i . Th e v ertex v is giv en the set { 1 , 2 , . . . , k } . Let f b e a prop er v ertex-coloring of H ′ assigning to eac h vertex a color from its set. Denote f ( v ) = i , then f restricted to G i is a prop er vertex-c oloring of G assigning to eac h v ertex w ∈ V a color in S(w). ✷ Lemma 6.2 BIP AR TI TE GRAPH 3 -CHOO SAB ILI TY is Π p 2 -c omplete. 25 Pro of It is easy to see that BG 3 -CH ∈ Π p 2 . W e tr an s form BG (2 , 3) -CH to BG 3 -CH . Let G = ( V , E ) and f : V 7→ { 2 , 3 } be an in s t ance of BG (2 , 3) -CH . W e shall construct a bip a rtite graph W such th at W is 3-c ho osable if and only if G is f -c ho osa b le . Let H b e the disj oint union of the graph s { G i,j : 1 ≤ i, j ≤ 3 } , wh ere eac h G i,j is a cop y of G . Let ( X, Y ) b e a b ipartiti on of the b ipartite graph H . Th e graph W is obtained from H by adding t w o new ve rtices u and v , joining u to eve ry v ertex w ∈ X for wh ich f ( w ) = 2, and j o inin g v to ev ery vertex w ∈ Y for w hic h f ( w ) = 2. Since H is bipartite, W is also a b ipartit e graph. It is easy to see that if G is f -c ho osa b le, then W is 3-c ho osable. W e now prov e that if W is 3-c ho osable, then G is f -c ho osable. F or ev ery w ∈ V , let S ( w ) b e a set of size f ( w ), s u c h that S ( w ) ∩ { 1 , 2 , 3 } = ∅ . F or eve ry i and j , 1 ≤ i, j ≤ 3, on the vertices of the graph G i,j w e p ut the sets S ( w ) with the ve rtices for which f is equal to 2 receiving another color as follo ws: to the vertic es w hic h b elong to X w e add the color i , whereas to the v ertice s whic h belong to Y w e add the color j . Th e ve rtices u and v are b oth g iven the set { 1 , 2 , 3 } . Let f b e a p roper ve rtex-coloring of H ′ assigning to eac h ve rtex a color from its set. Denote f ( u ) = i and f ( v ) = j , then f restricted to G i,j is a p roper vertex-co loring of G assigning to eac h v ertex w ∈ V a color in S ( w ). ✷ Pro of of Theorem 1.18 The pro of is b y induction on k . F or k = 3, the result follo ws from lemma 6.2. Assu m ing that the result is true f o r k , k ≥ 3, we pro ve it is true for k + 1. It is easy to see that BG ( k + 1) -CH ∈ Π p 2 . W e transform BG k -CH to BG ( k + 1) -CH . Let G = ( V , E ) b e an instance of BG k -CH . W e shall constr u ct a bipartite graph W suc h that W is ( k + 1)-c ho osable if and only if G is k -c ho osable. Let H b e the disjoint u nion of the graph s { G i,j : 1 ≤ i, j ≤ ( k + 1) 2 } , where eac h G i,j is a copy of G . Let ( X, Y ) b e a bipartition of the bipartite graph H . The graph W is obtained from H b y adding t wo new v ertices u and v , joining u to ev ery v ertex of X , and j o inin g v to ev ery v ertex of Y . It is easy to see that if G is k -c ho o sable, th e n W is ( k + 1)-c ho osable. In a similar w a y to the pro of of lemma 6.2, we can p ro v e that if W is ( k + 1)-c ho osable, then G is k -c ho osable. ✷ 26 7 The strong c hoice n um b er Let G = ( V , E ) b e a graph, and let V 1 , . . . , V r b e p ai rw ise d isjoin t subsets of V . W e denote by [ G, V 1 , . . . , V r ] the graph obtained from G by add ing to it the un io n of cliques in d uces b y eac h V i , 1 ≤ i ≤ r . Supp ose th a t G = ( V , E ) is a graph with maxim um degree at most 1. W e claim that G is strongly k -c hoosable for ev ery k ≥ 2. T o see that, let V 1 , . . . , V r b e pairwise disjoint subsets of V , eac h of size at most k . The graph [ G, V 1 , . . . , V r ] has maxim u m degree at most k , and therefore b y corollary 1.9 it is k -c ho osable. Pro of of Theorem 1.19 Let G = ( V , E ) b e a strongly k -co lorable graph. Let V 1 , . . . , V r b e pairwise disjoin t s ubsets of V , eac h of s iz e at most k + 1. Without loss of generalit y , we can assum e that V 1 , . . . , V m are subsets of size exactly k + 1, and V m +1 , . . . , V r are subsets of size less than k + 1. Let H b e the graph [ G, V 1 , . . . , V r ]. T o complete th e p roof, it su ffices to sh o w th a t H is ( k + 1)-colorable. F or ev ery i , 1 ≤ i ≤ m , we define W i = V i − { c } f or an arbitrary elemen t c ∈ V i , whereas for every j , m + 1 ≤ j ≤ r , we defi n e W i = V i . Sin c e [ G, W 1 , . . . , W r ] is k -co lorable, there exists an indep endent set S of H w hic h is comp osed of exactly one v ertex from eac h V i , 1 ≤ i ≤ m . F or every i , 1 ≤ i ≤ m , we d e fi n e W i = V i − S , whereas for ev ery j , m + 1 ≤ j ≤ r , we defin e W i = V i . Since [ G, W 1 , . . . , W r ] is k -co lorable, we can obtain a p roper ( k + 1)-v ertex coloring of H b y u s ing k colors for V − S and another color f o r S . ✷ Lemma 7.1 Supp ose that k , l ≥ 1 . If F is a family of k + l sets of size k + l , then it is p ossible to p ar tition F into a family F 1 of k sets and a family F 2 of l sets, to cho ose for e ach set S ∈ F 1 a su bset S ′ ⊆ S of size k , and to cho ose for e ach set T ∈ F 2 a su bset T ′ ⊆ T of size l , so tha t S ′ ∩ T ′ = ∅ for every S ∈ F 1 and T ∈ F 2 . Pro of Supp ose that F = { C 1 , . . . , C k + l } , and defin e C = ∪ k + l i =1 C i . F or ev ery partition π of C in to the tw o subs e ts A and B , we denote R ( π ) = { V ∈ F : | V ∩ A | > k } , L ( π ) = { V ∈ F : | V ∩ B | > l } , and M ( π ) = { V ∈ F : | V ∩ A | = k and | V ∩ B | = l } . W e no w start with the partition of C in to the tw o s ubsets A = C and B = ∅ , and start mo ving one element at a time from A to B unt il w e obtain a partition π 1 of C int o the t wo subsets A and B and a partition π 2 in to the t wo subsets A ′ = A − { c } and B ′ = B ∪ { c } , such that |R ( π 1 ) | > k and |R ( π 2 ) | ≤ k . It is easy to that 27 L ( π 2 ) ⊆ L ( π 1 ) ∪ M ( π 1 ), and therefore |L ( π 2 ) | < l . W e no w partition M ( π 2 ) into t wo subs e ts M 1 and M 2 , su c h that F 1 = R ( π 2 ) ∪ M 1 has size k and F 2 = L ( π 2 ) ∪ M 2 has size l . F or eve ry set S ∈ F 1 w e c h o ose a subset S ′ ⊆ S ∩ A ′ of size k , whereas for every T ∈ F 2 w e c h o ose a subset T ′ ⊆ T ∩ B ′ of size l . Since A ′ and B ′ are d isjoin t, w e ha ve that S ′ ∩ T ′ = ∅ f o r every S ∈ F 1 and T ∈ F 2 . ✷ Lemma 7.2 Supp ose th at k , m ≥ 1 . If F is a f amily of km sets of size k m , then i t is p ossible to p artition F into the m subsets F 1 , . . . , F m , e ach of size k , and to cho ose f o r e ach set S ∈ F a subset S ′ ⊆ S of size k , so that S ′ ∩ T ′ = ∅ for every i 6 = j , S ∈ F i and T ∈ F j . Pro of By indu c tion on m . F or m = 1 the result is trivial. Assumin g th a t the result is true for m , m ≥ 1, w e p ro v e it is true for m + 1. Let F b e a f a mily of k ( m + 1) sets of size k ( m + 1). By lemma 7.1, it is p ossible to p artit ion F into a family F 1 of k sets and a family F 2 of k m sets, to c ho ose for eac h S ∈ F 1 a su b set S ′ ⊆ S of size k , and to c ho ose for eac h set T ∈ F 2 a subset T ′ ⊆ T of s iz e k m , so that S ′ ∩ T ′ = ∅ for ev ery S ∈ F 1 and T ∈ F 2 . The p roof is completed by applying the indu ct ion hyp othesis on F 2 . ✷ Pro of of Theorem 1.20 Let G = ( V , E ) b e a s t ron gly k -c ho osable graph. Let V 1 , . . . , V r b e pairwise disjoint sub sets of V , eac h of size at most k m . Let H b e the graph [ G, V 1 , . . . , V r ]. T o complete the pro of, it su ffi c es to sh o w that H is k m -c ho osa b le. F or eac h v ∈ V , let S ( v ) b e a set of size k m . By lemma 7.2, for ev ery i , 1 ≤ i ≤ r , is it p ossible to p a rtition V i in to the m subsets V i, 1 , . . . , V i,m , eac h of size at most k , an d to c h o ose for eac h ve rtex v ∈ V i a subset C ( v ) ⊆ S ( v ) of size k , so that C ( u ) ∩ C ( v ) = ∅ for ev ery p 6 = q , u ∈ V i,p and v ∈ V i,q . Since the graph [ G, V 1 , 1 , . . . , V r,m ] is k -c ho osable, w e can obta in a prop er vertex-co loring of H assigning to eac h v ertex a color from its s et. ✷ Pro of of Theorem 1.21 Apply lemma 7.2 as in pro of of th eorem 1.20. ✷ Pro of of Corollary 1.22 If is pro ved in [9] that if G is a 4-regular graph on 3 n vertice s and G has a decomp osition in to a Hamiltonian circuit and n pairwise ve rtex disjoint triangles, then ch ( G ) = 3. The result follo ws fr om theorem 1.21. ✷ Pro of of Theorem 1.23 Sin c e sχ (1) = 2, we can assume that d > 1. Sup pose first th a t d is ev en, and denote d = 2 r . Construct a graph G with 12 r − 3 v ertices, partitioned into 8 classes, as follo ws. Let these classes b e A, B 1 , B 2 , C 1 , C 2 , D 1 , D 2 , E , where | A | = | D 1 | = | D 2 | = 2 r , | B 1 | = | B 2 | = r , 28 | C 1 | = | C 2 | = r − 1, a nd | E | = 2 r − 1. Eac h vertex in A is joined by edges to ea ch mem b er of B 1 and eac h m e mb e r of B 2 . Eac h memb er of D 1 is adjacen t to eac h mem b er of D 2 . Consider the follo wing partition of the s e t of v ertices of G into three classes of cardinalit y 4 r − 1 eac h: V 1 = B 1 ∪ C 1 ∪ D 1 , V 2 = B 2 ∪ C 2 ∪ D 2 , V 3 = A ∪ E . W e claim that H = [ G, V 1 , V 2 , V 3 ] is not (4 r − 1)-colo rab le. In a p roper (4 r − 1)-v ertex coloring of H , ev ery color used for coloring th e vertic es of A m ust app ear on a vertex of C 1 ∪ D 1 and on a v ertex of C 2 ∪ D 2 . Sin c e | C 1 ∪ C 2 | < | A | , there is a color used for coloring the vertic es of A which app ears o n b oth D 1 and D 2 . But this is imp ossible as ea ch v ertex in D 1 is adjacen t to eac h m emb er of D 2 . Thus sχ ( G ) > 4 r − 1 and as th e maximum d eg ree in G is 2 r , this sh ows that sχ (2 r ) ≥ 4 r . Supp ose next that d is o dd, an d denote d = 2 r + 1. Construct a graph G with 12 r + 3 ve rtices, partitioned in to 8 classes, a s fol lo ws . Let these classes b e n a med as b efore, w here | A | = | D 1 | = | D 2 | = 2 r + 1, | B 1 | = r + 1, | C 1 | = r − 1, | B 2 | = | C 2 | = r , and | E | = 2 r . In the same m a nn er w e can prov e that [ G, V 1 , V 2 , V 3 ] is not (4 r + 1)-colorable. Thus sχ ( G ) > 4 r + 1 and as the maxim um degree in G is 2 r + 1, this sh o ws th a t sχ (2 r + 1) ≥ 4 r + 2, completing the pro of. ✷ References [1] N. Alon, The str ong chr omatic numb er of a gr aph , Random Stru ctur es and Algorithms 3 (1992 ), 1-7. [2] N. Alon, Choic e numb ers of gr aph s; a pr ob abilistic appr o ach , Com b in a torics, Pr o bab ility and Computing, in pr ess. [3] N. Alon and J. H. S pencer, The Probabilistic Metho d , Wiley , 1991. [4] N. Alon and M. T arsi, Colorings and orientations of gr aphs , Combinatorica , in press. [5] C. Berge, Graphs and H ypergraphs , Dun od, Paris, 1970. [6] B. Bollob´ as, The c hr omatic nu m b er of r andom gr aphs , Combinatoric a 8 (1988), 49-55. [7] B. 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