Properly Coloured Cycles and Paths: Results and Open Problems

Prop erly Coloured Cycles and P aths: Results and Op en Problems Gregory Gutin ∗ Eun Jung Kim † Abstract In this pap er, we consider a num b er of results and seven conjec- tures on prop erly edge-colour ed (PC) paths and cycles in edge-colo ured m ultigraphs. W e ov erv iew so m e kno wn r esults and prov e new ones. In particular, w e consider a family of transfor m ations of an edge-co loured m ultigraph G into a n ordinary g raph that a llo w us to chec k the ex- istence PC cycles and PC ( s, t )-paths in G and, if they exist, to find shortest ones among them. W e raise a pr oblem of finding the optimal transformatio n and consider a p ossible solutio n to the problem. 1 In tro duction The class of ed g e-coloured multigraphs generalize directe d graphs. There are several other generalizat ions of d irec ted graph s suc h as arc-coloured digraphs, h yp ertournaments a nd star hyp e rgraphs, but the class of edge- coloured m ultigraphs has b een giv en the m ain atten tion in g raph theo ry literature b ecause man y concepts and results on directed graphs can b e extended to edge-coloured multig raphs and there are s everal imp ortant ap- plications of edge-coloured m u lti graphs. F or a m o re extensive treatmen t of this topic, see [6, 7]. In this pap er w e o v erview some known resu l ts on prop erly coloured (PC) cycles and paths in edge-col oured multigraphs, pr o ve new ones and consider sev eral op en pr o blems on th e topic. In S e ction 2 w e br ie fly consider a prob- lem of whether an edge-coloured graph has a PC cycle. In Sections 3 and 4, w e offer a usefu l to ol to study edge-coloured m u lt igraphs. In inv estigating problems on PC subgraphs of edge- coloured multigraphs, it is con venien t to transform an edge-coloured graph into a n ordinary graph . W e suggest a new tec hnique that somewhat automates this transformation. Moreo v er, ∗ Department of Computer Science, Roy al Hollo w a y , Un i versi ty of London, Egham, Surrey TW20 0EX, UK, gutin@cs .rhul.ac. uk † Department of Computer Science, Roy al Hollo w a y , Un i versi ty of London, Egham, Surrey TW20 0EX, UK, eunjung@ cs.rhul.a c.uk 1 b y pr o ving some new r e sults, w e illus tr a te ho w the prop osed tec hniqu e al- lo ws us to ob tain more efficient algorithms for PC cycle and PC ( s, t )-path problems by redu ci ng the order an d size of the transformed graph . W e raise a p r o blem of determinin g the min imum order and size of the transformed graph, and d e scrib e the family of graphs that ma y b e the solution to t he problem. In Section 5 w e study long PC cycles and p a ths in arbitrary edge-colo ured m u lt igraphs and Section 6 is d e v oted to longest (mostly Hamilton) PC cycles in edge-colo ured complete graphs. An m -path-cycle subgraph F o f a m u ltigraph G is a vertex-disjoin t union of m paths and a n umb er of cycles in G (some cycles can b e of length 2). If m = 0, we call F a cycle subgraph of G . F or a v ertex set X o f a m u lt igraph G , G h X i d e notes th e subgraph of G induced by X . F or a p a ir s, t of distinct vertic es of G , a path b et ween s and t is called an ( s, t )- path . W e consider edge-coloured m ultigraphs , i.e., undirected multigraphs in whic h eac h edge has a colour, but no parallel edges h a ve th e same colour. If an ed g e-colo ured multigraph G has c colours, we assume that the colours are 1 , 2 , . . . , c and w e call G a c - edge-co loured multigraph. W e d e note the colour of an edge e of an edge-colo ured m ultigraph G by χ ( e ) . When G has no parallel edges, w e call G an edge-coloured graph . Let G b e a c -edge-coloured m u l tigraph and let v ∈ V ( G ). By N i ( v ) we denote the set of neighbour s of v adjacen t to v by an edge of colour i ; let d i ( x ) = | N i ( x ) | . The maximum ( minim um ) mono c hromatic degree of G = ( V , E ) is defi n ed b y ∆ mon ( G ) = max { d j ( v ) : v ∈ V , 1 ≤ j ≤ c } ( δ mon ( G ) = min { d j ( v ) : v ∈ V , 1 ≤ j ≤ c } ) . Let χ ( v ) = { i : 1 ≤ i ≤ c, N i ( v ) 6 = ∅} . A path or cycle Q of G is prop erly coloured (PC) if ev ery t wo adjacen t edges of Q are of differen t colours. 2 Existence of PC Cycles Since a pair of parallel edges in a c -edge-coloured multigraph ( c ≥ 2) f o rms a PC cycle, in this section, we consider only c -edge-coloured graphs. It is easy to see that the problem of c hecking whether a c -edge-coloured graph has a PC cycle is more ge neral (ev en for c = 2) than the simple problem of v erifying whether a d igraph cont ains a directed cycle. Indeed, consider a digraph D and, to obtain a 2-edge-coloured graph G from D , replace eac h arc xy of D with edges xz xy and z xy y of colours 1 and 2, where z xy is a new v ertex ( z xy 6 = z x ′ y ′ pro vided xy 6 = x ′ y ′ ). Observe that G has a PC cycle if and only if D has a directed cycle. 2 The follo wing theorem by Y eo [19] provides a s i mple recursiv e wa y of c hecking whether a c -edge-coloured graph has a PC cycle. (F or c = 2, Theorem 2.1 w as first p ro v ed b y Grossman and H¨ aggkvist [12].) Theorem 2.1. L et G b e a c -e dge- c olour e d gr aph, c ≥ 2 , with no PC cycle. Then, G has a v er tex z ∈ V ( G ) su ch that no c onne cte d c omp onent of G − z is joine d to z with e dges of mor e than one c olour. Let us consider the follo wing fu nctio n in tr o duced by Gutin [13]: d ( n, c ), the minimum num b er k such that ev ery c -edge-coloured graph of order n and minim um mono c h romat ic degree at least k has a PC cycle. It was prov ed in [13] that d ( n, c ) exists and that d ( n, c ) ≤ 1 ⌊ c/ 2 ⌋ (log 2 n − 1 3 log 2 log 2 n + Θ(1)) . (1) Ab ouelao ualim et al. [1] stated a conjecture which implies that d ( n, c ) = 1 for eac h c ≥ 2. Using a r ecur siv e constru c tion in spired by T heo rem 2.1 of c -edge-co loured graphs with minim um mono c hromatic degree p and without PC cycles, Gutin [13] show ed that d ( n, c ) ≥ 1 c (log c n − log c log c n ) (2) and, th u s, the conjecture do es not hold. Th e b ounds (1) and (2) imply that d ( n, c ) = Θ(log 2 n ) f o r every fi x ed c ≥ 2. Conjecture 2.2. [13] Ther e is a f u nctio n s ( c ) dep endent only on c such that d ( n, c ) = s ( c ) log 2 n (1 + o (1)) . In particular, it wo uld b e inte resting to determine s (2). 3 P-Gadgets W e consider gadget constructions which generalize some kno wn construc- tions mentio ned b elo w. The P-gadget graphs G ∗ and G ∗∗ of an edge-coloured m u lt igraph G describ ed in the next section allo w one to transform sev eral problems on prop erly coloured su bgraphs of G into p erfect matc hing p rob- lems in G ∗ or G ∗∗ . Let G b e an edge-colo ured multigraph and let G ′ = G − { x ∈ V ( G ) : | χ ( x ) | = 1 } . F or eac h x ∈ V ( G ′ ) let G x b e an arbitrary (non-edge-coloured) graph with the follo wing four prop erties: P1 { x q : q ∈ χ ( x ) } ⊆ V ( G x ); P2 G x has a p erfect matc hin g; 3 P3 F or eac h p 6 = q ∈ χ ( x ), if the graph G x − { x p , x q } is not empt y , it h as a p erfect matc hin g; P4 F or eac h set L ⊆ χ ( x ) with at least 3 elements; if the graph G x − { x l : l ∈ L } is not empt y , it has no p erfect matc hing. Eac h G x with the prop erties P1-P4 is called a P-gadget . L e t us consider the follo wing thr e e P-gadgets; the first t wo are kno w n in the literature and the third one is new. 1. One P-gadget is due to Szeider [17]: V ( G x ) = { x i , x ′ i : i ∈ χ ( x ) } ∪ { x ′′ a , x ′′ b } and E ( G x ) = { x ′′ a x ′′ b , x ′ i x ′′ a , x ′ i x ′′ b : i ∈ χ ( x ) } ∪ { x i x ′ i : i ∈ χ ( x ) } . W e will call this the SP-gadget . 2. Another gadget is due to Bang-Jensen and Gutin [4]: V ( G x ) = { x j : j ∈ χ ( x ) } ∪ { y j : j ∈ χ ( x ) \ { m, M }} , where m = min χ ( x ) , M = max χ ( x ), and E ( G x ) = { x j y k : j ∈ χ ( x ) , k ∈ χ ( x ) \{ m, M }}∪{ x j x k : j 6 = k ∈ χ ( x ) } . W e will call this the BJGP-gadget . 3. The follo w in g new gadget is a sort of crossov er of the ab o ve t w o and is called the XP-gadget : V ( G x ) = { x j : j ∈ χ ( x ) } ∪ { y j : j ∈ χ ( x ) \ { m, M }} , where m and M are defin ed ab o v e, and E ( G x ) = { x m x M } ∪ { x j y j , x m y j , x M y j : j ∈ χ ( x ) \ { m, M }} . It is not difficult to verify that the tree P-gadgets ind eed satisfy P1-P4. Let z = χ ( x ). Observ e that the SP - gadget h a s 2 z + 2 vertice s and 3 z + 1 edges, the BJGP-gadget 2 z − 2 v er tices and z (3 z − 5) / 2 edges, the XP-gadget 2 z − 2 ve rtices and 3 z − 5 edges. Th us, the XP-gadget has th e m i nim um n um b er of v ertices and edges among the th ree P-gadgets. It is not difficu lt to v erify that the XP-gadget has the m i nim um n u m b er of vertices and edges among all p ossible P-gadgets for z = 2 , 3 , 4. P erh a ps, th i s is true for an y z . Conjecture 3.1. The XP-gadget has the minimum numb er of vertic es and e dges among al l p ossible P-gadgets for every z ≥ 2 . W e will see in the next section why minimizing the num b ers of vertic es and edges in P-gadgets is imp ortan t for sp eeding up some algorithms on edge-colo ured m ultigraphs. 4 4 P-gadget Graph s Let G b e a c -edge-coloured m ultigraph and let G x b e a P-gadget f or x ∈ V ( G ′ ). The grap h G ∗ is d efined as follo w s: V ( G ∗ ) = ∪ x ∈ V ( G ′ ) V ( G x ) and E ( G ∗ ) = E 1 ∪ E 2 , wh e re E 1 = ∪ x ∈ V ( G ′ ) E ( G x ) and E 2 = { y q z q : y , z ∈ V ( G ′ ) , y z ∈ E ( G ) , χ ( y z ) = q , 1 ≤ q ≤ c } . Let s, t b e a pair of d i stinct vertices of G and let H = G − { s, t } . Let G ∗∗ b e constructed fr o m H ∗ b y add i ng s and t and edges E 3 = { sx i : sx ∈ E ( G ) , χ ( sx ) = i } ∪ { tx i : tx ∈ E ( G ) , χ ( tx ) = i } . W e will denote the num b er of v ertices and edges in m ultigraph s G , G ∗ and G ∗∗ b y n, m, n ∗ , m ∗ , n ∗∗ and m ∗∗ , resp ectiv ely . The follo wing result relates p erfect matc hings of G ∗ with PC cycle su b- graphs of G . PC cycle sub g raphs are imp ortan t in sev eral p roblems on edge-colo ured m ultigraphs (for example, for the PC Hamilton cycle pr o b- lem), see [6]. Recall that G ′ = G − { x ∈ V ( G ) : | χ ( x ) | = 1 } . Theorem 4.1. L et G b e a c onne cte d e dge-c olour e d multigr aph such that G ′ is non-empty. Then G has a PC cycle sub gr aph with r e dges if and only if G ∗ has a p erfe ct matching with exactly r e dges in E 2 . Pro o f: Let M b e a p erfect matc hing of G ∗ with exactly edges x 1 p 1 y 1 q 1 , . . . , x r p r y r q r in E 2 . F or a v ertex x of G ′ , let Q x b e the set of edges in E 2 adjacen t to G x . By P2, eac h G x has even num b er of v ertices ( x ∈ V ( G ′ )) an d s in c e M is a p erfect matc hin g in G ∗ , there is even num b er of ed ges in Q x . By P4, Q x has either no edges or t wo edges for eac h x ∈ V ( G ′ ). Let X b e th e set of all v ertices x ∈ V ( G ′ ) suc h that | Q x | = 2. Then, by th e d efi niti on of G ∗ , G h X i con tains a PC cycle factor. It remains to observ e that | X | = r . No w let F b e a PC cycle sub grap h of G w i th r edges. Observe that the edges of F corresp ond to a set Q of r ind e p enden t edges of G ∗ and that either no edges or t wo edges of Q are adjacen t to G x for eac h x ∈ V ( G ′ ). No w d e lete the vertice s adjacen t with Q fr o m eac h G x and observ e that eac h remaining non-empty gadget h a s a p erfect matc hing by P2 and P3. Com b i ning the p erfect m a tc hings of the non-empty gadgets with Q , we get a p erfect matc hin g of G ∗ with exactly r edges from E 2 . The fir st part of th e next assertion generalizes a result from [4]. The second part is b a sed on an approac h which leads to a more efficien t algo r it hm than in [2]. Corollary 4.2. One c an che ck whether an e dge-c olour e d multigr aph G has a PC c y cle and, if it do es, find a maximum PC cycle sub gr aph of G in time O ( n ∗ · ( m ∗ + n ∗ log n ∗ )) . Mor e over one c an find a shortest PC cycle in G in time O ( n · n ∗ · ( m ∗ + n ∗ log n ∗ )) . 5 Pro o f: W e ma y assume that G is connected and that G ′ is n ot empt y . By Theorem 4.1, it is enough to fin d a p erfect matc hing of G ∗ con taining the maxim um n u m b er of edges fr o m E 2 . Assign weigh t 0 (1, resp ec tiv ely) to edges of G ∗ in E 1 ( E 2 , r e sp ectiv ely). Now we need to find a maximum weigh t p erfect matc h ing of G ∗ whic h can b e d on e in time O ( n ∗ · ( m ∗ + n ∗ log n ∗ )) b y a matc hing algorithm in [11]. T o find a shortest PC cycle in G , c ho ose a v ertex x ∈ V ( G ′ ). W e will find a sh o rtest PC cycle in G tra versing x . By Th eo rem 4.1, it is enough to find a p erfect matc h i ng of G ∗ con taining the minim u m num b er of edges from E 2 while con taining at least one edge from E 2 so that th e corresp onding PC cycle in G should b e n o n-trivial. W e defin e the w eights on ed g es of G ∗ as follo ws. Assign M , w here M is a sufficiently large num b er, to eac h edge in E 2 inciden t with G x . F or all other edges, assign weigh t 1 (0, resp ectiv ely) to edges of G ∗ in E 1 ( E 2 , r e sp ectiv ely). A m a xim um w eigh t p erfect matc hin g of G ∗ con tains exac tly t w o edges of wei gh t M by P4, and con tains the minimum n um b er of edges in E 2 . Findin g a maximum w eigh t p erfect matc hing of G ∗ can b e done in time O ( n ∗ · ( m ∗ + n ∗ log n ∗ )) an d we iterate the pro cess for eac h x ∈ V ( G ′ ). The pro of of the follo wing result is analogous to the p roof of Theorem 4.1. Theorem 4.3. L et G b e an e dge- c olour e d multigr aph and let s, t b e a p air of distinct ve rtic es of G . If G ∗∗ is non-empty, then G has a PC 1-p ath-cycle sub gr aph with r e dges in which the p ath is b etwe en s and t if and only if G ∗∗ has a p erfe ct matching with exactly r e dges not in E 1 . The next assertion generalizes a result from [2]. Corollary 4.4. L et G b e an e dge-c olour e d multigr aph. One c an che ck whether ther e is a PC ( s, t ) - p ath in G in time O ( m ∗∗ ) and if G has one, a shortest PC ( s, t ) - p ath c an b e found in time O ( n ∗∗ · ( m ∗∗ + n ∗∗ log n ∗∗ )) . Pro o f: Let L b e a graph. Giv en a matc hing M in L , a path P in L is M − augmen ting if, for any p a ir of adjacen t ed g es in P , exactly one of them b elongs to M and the fi rst and last edges of P do n o t b elong to M . Consider a p erfect matc hing M of H ∗ , where H = G − { s, t } , whic h is a collect ion of p erfect matc h i ngs of G x for all x ∈ V ( G ′ ). The existence of a p erfect matc h ing in G x is guarante ed by P2. Observ e that G h a s a PC ( s, t )-path if and only if there is an M − augmen tin g ( s, t )-path P in G ∗∗ . Since an M − a ugmen ting path P can b e found in time O ( m ∗∗ ) (see [18]), we can find a PC ( s , t )-path in G , if one exists, in time O ( m ∗∗ ). T o find a shortest PC ( s, t )-path, w e assign eac h edge in S x ∈ V ( G ′ ) E ( G x ) w eight 0 and eve ry other edge of G ∗∗ w eight 1. Observ e that a minimum 6 w eight p erfect m atching Q in the w eigh ted G ∗∗ corresp onds to a shortest PC ( s, t )-path. Find ing a minimum weigh t p erfect matc hing can b e done in time O ( n ∗∗ · ( m ∗∗ + n ∗∗ log n ∗∗ )). 5 Long PC Cycles and P aths The follo win g interesti ng result and conjecture w ere obtained by Ab oue- laoualim, Das, F ernandez de la V ega, K a rpinski, Manouss a kis, Martinhon and Saad [1]. Theorem 5.1. [1] L et G b e a c -e dge-c olour e d multigr aph G with n vertic es and with δ mon ( G ) ≥ ⌈ n +1 2 ⌉ . If c ≥ 3 or c = 2 and n is even, then G has a Hamilton PC cycle. If c = 2 and n is o dd, then G has a PC cycle of length n − 1 . Conjecture 5.2. [1] The or em 5.1 holds if we r eplac e δ mon ( G ) ≥ ⌈ n +1 2 ⌉ b y δ mon ( G ) ≥ ⌈ n 2 ⌉ . W e cann ot replace δ mon ( G ) ≥ ⌈ n +1 2 ⌉ by δ mon ( G ) ≥ ⌈ n − 1 2 ⌉ due to th e follo wing example. Let H 1 and H 2 b e c -e dge-coloured complete m ultigraphs (for eac h p a ir x, y of v ertices and eac h i ∈ { 1 , 2 , . . . , c } and j ∈ { 1 , 2 } , H j has a edge b et w een x and y of colour i ) of order p + 1 that h a ve precisely one v ertex in common. Clearly , a longest P C cycle in H 1 ∪ H 2 is of length p + 1. Since the longest PC path problem is N P -hard, it makes sense to study lo wer b ounds on the length of a longest PC p ath. The f o llo wing result w as pro v ed b y Ab ouelaoualim et al. [1]. Theorem 5.3. L et G b e a c -e dge-c olour e d gr aph of or der n with δ mon ( G ) = d ≥ 1 . Then G has a PC p ath of length at le ast min { n − 1 , 2 ⌊ c 2 ⌋ d } . The authors of [1] raised the follo wing t w o conjectures. Conjecture 5.4. L e t G b e a c -e dge-c olour e d gr aph of or der n and let d = δ mon ( G ) ≥ 1 . Then G has a PC p ath of length at le ast min { n − 1 , 2 cd } . They also conjectured the follo win g analog of Theorem 5.3 for multi- graphs: Conjecture 5 .5. L et G b e a c -e dge- c olour e d multigr aph of or der n with δ mon ( G ) = d ≥ 1 . Then G has a PC p ath of length at le ast min { n − 1 , 2 d } . 6 Longest PC Cycles and Pa ths in Edge-Coloured Complete Graphs Let K c n denote a c -edge-coloured complete graph with n ve rtices. F eng, Giesen, Guo, Gu t in, Jensen and Rafiey [10] pr ov ed the follo wing: 7 Theorem 6.1. A K c n ( c ≥ 2 ) has a PC Hamilton p ath if and only if K c n c ontains a P C sp anning 1-p ath-cycle sub gr aph. This theorem w as fi rst pro ved b y Bang-Jensen and Gutin [4] for the case c = 2 and they conjectured that Theorem 6.1 holds for eac h c ≥ 2. Theorem 6.1 implies that the maximum order of a PC path in K c n equals the maxim u m order of a P C 1-path-cycle subgraph of K c n . As a result, the problem of find in g a longest PC p ath in K c n is p olynomial- time solv able f o r arb i trary c ≥ 2. T o see that a PC 1-path-cycle subgraph of K c n can b e found in p olynomial time, ad d a pair x, y of n e w vertice s to K c n together with all edges needed to hav e a complete multigraph on n + 2 v ertices. Let the colo ur of all edges b et wee n x and y and K c n b e c + 1 and let the colour of xy b e c + 2 . O b serv e that the maxim um ord e r of a PC 1-path- cycle su bgraph of K c n equals the maximum order of a PC cycle subgrap h of th e c + 2-edge- coloured complete graph describ ed ab o ve. It remains to apply Corollary 4.2. The p roblem of finding a longest P C cycle K c n has n o t b een solv ed yet for c ≥ 3 as we will see b elo w. F or c = 2, Saad [15] found a c haracterizatio n for longest PC cycles using the follo wing notions. A pair of distinct v ertices x, y of G are colour-connected if there exist PC ( x, y )-paths P and Q suc h that χ ( f P ) 6 = χ ( f Q ) and χ ( ℓ P ) 6 = χ ( ℓ Q ), where f P and f Q are the first edges of P and Q , resp ectiv ely , and ℓ P and ℓ Q are the last edges of P and Q , resp ectiv ely . W e s ay that G is colour-connected if ev er y pair of distinct v ertices of G is colour-connected. Saad’s c haracterization is as follo ws. Theorem 6.2. The length of a longest PC cycle in a c olour-c onne cte d K 2 n is e qual to the maximum or der of a PC cycle sub gr aph of K 2 n . Colour-connectivit y for K c n is an an equiv alence relation (see [6]). Usin g Theorem 6.2, Saad [15] sh o wed that the pr o blem of findin g a longest PC cycle in K 2 n is r a ndom p olynomial. Using a sp ecial case of Corollary 4.2, Bang-Jensen and Gutin [5] p ro ved that the p roblem is, in fact, p olynomial- time solv able. Th e orem 6.2 imp lie s the follo wing: Corollary 6.3. [15] A K 2 n has a PC Hamilton cycle if and only if K 2 n is c olour-c onne cte d and c ontains a PC c ycle factor. There is another c haracterization of K 2 n with a PC Hamilton cycle d ue to Bankfalvi and Bankfalvi, see [6]. T he straigh tforward extension of Corollary 6.3 is not tru e for any c ≥ 3 [6]. In fact, no charact erization of K c n with a PC Hamilton cycle is kn o wn for an y fixed c ≥ 3 an d it is a very interesting problem to obtain su c h a c haracterization. Ev en the follo w i ng problem by Benk ouar, Manoussakis, Pa sc hos and Saad [8] is still op en. Problem 6.4. Determine the c omplexity of the P C H a milton cycle pr oblem for c -e dge-c olour e d c omplete gr aphs when c ≥ 3 . 8 W e conjecture that th e P C Hamilton cycle p roblem for c -edge-coloured complete graphs when c ≥ 3 is p olynomial-time solv able. In absence of c haracterization of K c n with a PC Hamilton cycle, sufficien t conditions are int erest. Manoussakis, Spyratos, T uza and V oigt [14] pr o ved the next result. Prop o sition 6.5. If c ≥ 1 2 ( n − 1)( n − 2) + 2 , then every K c n has a PC Hamilton cycle. Let ∆ mon ( K c n ) d e note the largest mono c hr o matic degree of K c n . Bollob´ as and Erd˝ os [9] p osed the follo w in g : Conjecture 6.6. Every K c n with ∆ mon ( K c n ) ≤ ⌊ n/ 2 ⌋ − 1 has a PC Hamilton cycle. Improvi ng some previous results on this conjecture, Shearer [16] show ed that if 7∆ mon ( K c n ) < n , then K c n has a PC Hamilton cycle. So far, the b est asymptotic estimate wa s obtained by Alon and Gutin [3]. Theorem 6.7. [3] F or every ǫ > 0 ther e e xi sts an n 0 = n 0 ( ǫ ) so that for e ach n > n 0 , every K c n satisfying ∆ mon ( K c n ) ≤ (1 − 1 √ 2 − ǫ ) n c ontains a PC Hamilton cycle. References [1] A. Ab ouelaoualim, K.Ch. Das, W. F er nandez de la V eg a, Y. Manoussakis, C.A. Martinhon, and R. Saa d, Cy c les and pa t hs in e d ge-colo r ed graphs with given degrees, Manuscript, 20 07. 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