Contractible Hamiltonian Cycles in Triangulated Surfaces

Con tractible Hamiltonian Cycles in T riangulated Surfaces Ashish Kumar Upadh ya y Departmen t of Mathematics Indian Institut e of T echnology P at na P atliputra Colony , Patna 800 013, India. upadhy a y@iitp.ac.in Ma y 14, 2018 Abstract A triangulation of a surface is called q -equiv elar if eac h of its v ertices is inciden t with exactly q triangles. In 1 972 Altshuler had sho wn that an equiv elar triangulation of t orus has a Ha miltonian Circuit. Here w e presen t a necessary and sufficient condition for existence of a con tractible Hamiltonian Cycle in equiv elar triangulation of a sur face. AMS classification : 57Q15, 57M20, 57N05. Keyw ords : Contrac tible Hamiltonian cycles, Pr op er T r ees in Maps, Equive lar T riangulations. 1 In tro duction A graph G := ( V , E ) is without lo ops and suc h that no more than one edge jo ins tw o v ertices. A map on a surface S is an em b edding of a g raph G with finite num ber of v ertices suc h that the comp onents of S \ G a r e top ological 2 -cells. Thus , the closure of a cell in S \ G is a p − g onal disk, i.e. a 2- disk whose b oundary is a p − gon for some in teger p ≥ 3. A map is called { p, q } e quivelar if each verte x is inciden t with exactly q n um b ers of p -gons. If p = 3 then the map is called a q - equiv elar triangulatio n o r a degree - regular t r iangulation of t yp e q . A map is called a Simplicia l C omplex if eac h of its faces is a simplex. Th us a triangulatio n is a Simplic i a l C omplex F or a simplicial complex K, the graph consisting of the edges and v ertices of K is called the e dge-gr aph of K and is denoted b y E G ( K ). If X a nd Y are tw o simplicial complexes, t hen a (simplicial) isomorphism from X to Y is a bijection φ : V ( X ) → V ( Y ) suc h that for σ ⊂ V ( X ), σ is a simplex o f X 1 if and only if φ ( σ ) is a simplex of Y . Tw o simplicial complexes X and Y are called (simplicially) isomorphic (and is denoted by X ∼ = Y ) when suc h an isomorphism exists. W e iden tify t w o complexes if they are isomorphic. An isomorphism fro m a simplicial complex X to it self is called an auto mo r phism of X . All the auto morphisms of X form a group, whic h is denoted b y Aut ( X ). In 1 956, T utte [13] sho w ed that eve ry 4-connected planar graph has a Hamiltonian cycle. Later in 1970, Gr ¨ u n baum conjectured tha t ev ery 4-connected g r a ph whic h admits an em b edding in the torus has a Hamiltonian cycle. In the same article he also remarked that - probably t here is a function c ( k ) such that each c ( k )-connected graph of gen us at most k is Hamiltonian. In 1972, Duk e [8] sho w ed the existence of suc h a function and gav e an estimate [ 1 2 (5 + √ 16 k + 1)] ≤ c ( k ) ≤ { 3 + √ 6 k + 3 } where k ≥ 1. A. Altsh uler [1], [2] studied Hamiltonian cycles and paths in the edge graphs of equiv elar maps on the to rus. That is in the maps whic h are equiv elar of types { 3 , 6 } and { 4 , 4 } . He sho w ed that in the graph consisting of ve rtices and edges of equiv elar maps of ab ov e type t here exists a Hamiltonian cycle. He a lso sho w ed that a Hamilto nian cycle exists in ev ery 6-connected graph on the torus. In 1 998, Barnette [5] show ed that a n y 3 -connected graph other t ha n K 4 or K 5 con tains a contractible cycle o r contains a simple configur a tion as subgraphs. In this article we presen t a necessary and sufficien t condition fo r existence of a con tractible Hamiltonian cycle in edge graph of an equiv elar triangulatio n of surfaces. W e moreov er show that the contractible Hamiltonian cycle b ounds a tr ia ngulated 2-disk. If the equiv elar triangulation of a surface is on n ve rtices then this disk has exactly n − 2 triangles and all of its n ve rtices lie on the b oundary cycle. W e b egin with some definitions. 2 Definiti o ns and Preliminaries Definition 1 A p ath P in a gr aph G is a sub gr aph P : [ v 1 , v 2 , . . . , v n ] of G , such that the vertex set of P is V ( P ) = { v 1 , v 2 , . . . , v n } and v i v i +1 ar e e dges in P for 1 ≤ i ≤ n − 1 . Definition 2 A p ath P : [ v 1 , v 2 , . . . , v n ] in G is said to b e a c ycle if v n v 1 is also an e dge in P . Definition 3 A gr aph without any cycle s or lo ops is c al le d a tr e e If a surface S has an equiv elar triangulation o n n ve rtices then the pro o f of the Theorem 1 is g iven by considering a tree with n − 2 v ertices in the dual map of the degree-regular triangulatio n of the surface. W e define this tree a s f ollo ws : Definition 4 L et M denote a ma p on a surfac e S , whic h is the dual map o f a n vertex de gr e e-r e gular trian g ulation K of the surfac e. L et T denote a tr e e o n n − 2 vertic es on M . We sa y that T is a pr op er tr e e if : 2 1. whenev e r two vertic es u 1 and u 2 of T b elong to a fac e F in M , a p ath P [ u 1 , u 2 ] joining u 1 and u 2 in b oundary of F b elongs to T . 2. any p ath P in T wh i c h lies in a fac e F of M is of length at most q − 2 , whe r e M is a map of typ e { q , 3 } . If v is a v ertex of a simplicial complex X , then the num ber of edges con taining v is called the degree of v and is denoted b y deg X ( v ) (or deg( v )). If the num ber of i - simplices of a simplicial complex X is f i ( X ) (0 ≤ i ≤ 2), then the num ber χ ( X ) = f 0 ( X ) − f 1 ( X ) + f 2 ( X ) is called the Euler char acteristic of X . A simplicial complex is called neigh b ourly if eac h pair of its v ertices fo rm an edge. 3 Example : An equiv elar-triang ulation and its dual Non-Orien table degree-regular com binatorial 2-manifol d of χ = − 2 . ❅ ❅ ❅ ❈ ❈ ❈ ❈ ❈ ❈ ✄ ✄ ✄ ✄ ✄ ✄ ✑ ✑ ✑ ✑ ✑    ◗ ◗ ◗ ◗ ◗ ◗ ◗ ◗ ◗ ◗ ✑ ✑ ✑ ✑ ✑ ◗ ◗ ◗ ◗ ◗ ✄ ✄ ✄ ✄ ✄ ✄ ❈ ❈ ❈ ❈ ❈ ❈    ❅ ❅ ❅ ✑ ✑ ✑ ✑ ✑ ◗ ◗ ◗ ◗ ◗ ✑ ✑ ✑ ✑ ✑ ❇ ❇ ❇ ❇ ❇ ❆ ❆ ❆ ✁ ✁ ✁ P P P P P ❆ ❆ ❆ ❆ ❆ ❆ ❩ ❩ ❩ ❩ ❩ ❩ ✘ ✘ ✘ ✘ ✘ ✘ ❍ ❍ ❍ ✁ ✁ ✁ ✟ ✟ ✟ P P P P P ✁ ✁ ✁ ✟ ✟ ✟ ✟ ✟ ✟ ✟ ✟ ✟ P P P P P   ✥ ✥ ✥ ✥ ✥ ✥ ✥ ✥ ✁ ✁ ✁ ❆ ❆ ❆ ✁ ✁ ✁ ✁ ✁ ✁ ❆ ❆ ❆ ❆ ❆ ❆ 5 12 10 4 6 8 12 7 5 3 6 4 10 8 11 1 2 7 9 12 10 3 11 8 12 5 3 u 2 u 21 u 1 u 22 u 3 u 18 u 25 u 23 u 24 u 26 u 27 u 20 u 28 u 19 u 4 u 5 u 7 u 16 u 15 u 17 u 6 u 10 u 9 u 8 u 11 u 12 u 13 u 14 ❆ ❆ ❆ ✁ ✁ ✁ ✁ ✁ ✁ ❍ ❍ ❍ ❆ ❆ ❆ ✟ ✟ ✟ ❅ ❅ ❅      ❅ ❅ ❍ ❍ ❍ ✟ ✟ ✟ ✁ ✁ ✁ ❆ ❆ ❆ ❆ ❆ ❆ ✁ ✁ ✁ ❅ ❅   ✟ ✟ ✟ ❍ ❍ ❍    ❅ ❅ ❅ ✁ ✁ ✁ ✁ ✁ ❵ ❵ ❆ ❆ ❆ ❅ ❅ ✟ ✟ ✟ ❅ ❅      ❅ ❅ ❆ ❆ ❆ ✡ ✡ ✡ ❍ ❍ 3 4 Some facts ab out prop er tree s. Lemma 4.1 L et v ∈ V ( T ) b e a vertex in a pr op er tr e e T . Then deg( v ) ≤ 3 . Proo f : Let M denote the dual map of a triangulation K . Th us deg ( u ) = 3 for all u ∈ V ( M ). Since T is a subgraph of the edge graph of M , deg ( v ) ≤ 3 f o r all v ∈ V ( T ). ✷ Lemma 4.2 L et T b e a pr op er tr e e and m b e the numb er of vertic es of de gr e e 3 in T . Then the numb er of vertic es o f de gr e e one in T = m + 2 . Proo f : Let P 1 denote a path of maxim um length in T . Then P 1 has t w o ends whic h are also ends of T , for otherwise P 1 will not b e of maxim um length. If t here is no v ertex o f degree 3 in T whic h also lies in P 1 then P 1 = T , as T is connected, and we are done. Otherwise, let u 1 b e a v ertex o f degree 3 suc h t ha t u 1 ∈ P 1 ∩ T . Let u 1 b e the initia l p oin t of a path P 2 of maximum length in the tree T ′ = T \ P 1 . Th us P 2 is edge disjoin t with P 1 . Then b y the ab o v e arg ument the end o f P 2 other than u 1 is also an end of T ′ and hence of T . F urther, if there is a v ertex w 1 of degree 3 on P 2 ∩ T ′ , w e rep eat the a b o v e pro cess to find a n end of T . Thus , for eac h v ertex of degree 3 in T w e g et an end of T . This to gether with ends of P 1 pro v es that the n um b er of ends of T = m + 2. ✷ Lemma 4.3 L et T b e a pr op er tr e e in a p olyhe d r al ma p M of typ e { q , 3 } on a surfac e S . Then T T F 6 = ∅ for any fa c e F of M . Proo f : Let e denote the n um b er of v ertices o f degree one in T . Since T has n − 2 v ertices, it has n − 3 edges. W e claim that the n − 3 edges o f T lie in exactly n − e faces o f M . T o pro v e this we enume rate the num ber o f faces of M with whic h the edges of T are inciden t. W e construct sets E and ˜ F as follo ws. Let E b e a singleton set whic h con tains an edge e 1 of T and F 1 and F 2 b e the fa ces of M suc h that e 1 lies in them. Put ˜ F := { F 1 , F 2 } . Add an a djacen t edge e 2 of e 1 to E . There is exactly one face F 3 differen t fr om F 1 and F 2 suc h that e 2 lies in F 3 . Add this to set ˜ F t o obtain ˜ F := { F 1 , F 2 , F 3 } . Successiv ely , w e add edges to the set E whic h a re adj a cen t to edges in E till w e exhaust all the edges of T . Each a dditional edge added to E con tributes exactly one face to the set ˜ F unless it is adjacen t to t w o edges in the set E . Thus the n umber of faces in ˜ F = (n um b er of edges of T - n um b er of vertice s of degree three) + 1. In a 3- t r ee, the n um b er of v ertices of degree 3 = n umber of end p oin t - 2. Th us # ˜ F = n − 3 − ( e − 2) + 1. That is # ˜ F = n − e . Let F ( M ) denote the set o f all fa ces of M . Let G = F ( M ) \ ˜ F . Then # G = e . W e claim that a n end v ertex o f T lies on exactly one face F ∈ G . Observ e that each v ertex u of T is incident with exactly three distinct faces F 1 , F 2 and F 3 of M . The edge o f T inciden t with u lies in t wo of these fa ces, sa y F 1 and F 2 , i.e. , F 1 , F 2 ∈ ˜ F . Since, u is an end v ertex, there is no edge of T whic h is inciden t with F 3 , for otherw ise 4 this violat es the defin ition of T . Thus u is inciden t with exactly one face F 3 of M such that F 3 ∈ G . Since, u is an a rbitrary end p oint this h yp othesis holds f o r all the end v ertices. If it happ ens that for some end v ertices u 1 and u 2 of T , the corresp onding faces W 1 = W 2 ∈ G then w e w ould ha v e u 1 and u 2 on the same face of M but no path on W 1 joining u 1 and u 2 lies in T . This con tradicts the definition of T . Th us G has exactly e distinct elemen ts. This prov es t he lemma. ✷ Lemma 4.4 L et K b e a n vertex de gr e e r e gular triangulation of a surfac e S . L et M denote the dual p olyhe dr on c orr esp o n ding to K and T b e a n − 2 vertex pr op er tr e e in M . L et D de n ote the sub c ompl e x of K wh i c h is dual of T . T hen D is a triangulate d 2-disk and bd ( D ) is a Hamiltonian cycle in K . Proo f : By definition of a dual, D consists of n − 2 tria ng les corresp onding to n − 2 v ertices of T . Tw o triangles in D ha v e a n edge in common if the corres p onding v ertices are adjacent in T . It is easy to see that D is a collapsible simplicial complex and hence it is a triang ula ted 2-disk. Moreo v er, since T has v ertices of degree one, bd ( D ) 6 = ∅ , and b eing b oundary complex of a 2-disk it is a connected cycle. Observ e that the n um b er of edges in n − 2 triangles is 3( n − 2 ) and for eac h edge of T exactly 2 edges are iden tified. Henc e the n um b er of edges whic h remain uniden tified in D is 3( n − 2) − 2( n − 3) = n . Similarly the num ber of vertice s in bd ( D ): = ∂ D = n . If there are v ertices v 1 , v 2 ∈ ∂ D suc h that v 1 and v 2 lie on a path of length < n and v 1 = v 2 . This means there are faces F 1 and F 2 in D with v 1 ∈ F 1 , v 2 ∈ F 2 , F 1 6 = F 2 and F 1 not adjacen t to F 2 . Th us there exist a face F ′ in D suc h that the vertex u F ′ in T corr esp onding to F ′ , do es not b elong t o the face F ( v 1 ) corresponding to v ertex v 1 . But this con tradicts that T is a prop er tree. Th us a ll the cycle ∂ D contains exactly n distinct v ertices. Since # V ( K ) = n , ∂ D is a Hamilto nian cycle in K . ✷ Theorem 1 The e dge gr aph E G ( K ) of an e quivelar triangulation K of a surfac e has a c ontr actible Hamiltonian cycle if and only if the e dge gr aph of c orr esp onding dual map M of K has a pr op er tr e e. Proo f : The a b o v e Lemma 4.4 sho ws the if part. Conv ers ely , let K denote an equiv elar triangulation and H := ( v 1 , v 2 , v 3 , . . . , v n ) denote a con tractible Hamilto- nian cycle in E G ( K ). Let τ 1 , τ 2 , . . . , τ m denote the faces of triangulated disk whose b oundary is H . W e claim that all the triangles ha v e t heir v ertices on b oundary of the disk, i.e. on H . F or otherwise there will b e identifications on the surface b ecause a ll the v ertices of K also lie on H . If x denotes the n um b er of triangles in this disk then the Euler c haracteristic relation give s us 1 = n − [ (3 × x ) − n 2 + n ] + x . Th us, x = n − 2. So that m = n − 2. Now, in the edge graph o f dual map M of K , consider the gr a ph corresp onding to this disk whose v ertices corres p ond to the dual of fa ces τ 1 , τ 2 , . . . , τ m . No w it is easy to chec k that this graph is a tree whic h is also a prop er tree. ✷ Ac kno wledgemen t : The author thanks D. Ba r nette [4] for reading and a ppreciating the idea o f Prop er T ree. That this tr ee ma y b e a necess ary and sufficien t condition for existence of separating Hamiltonian cycle ( Theorem 1) w as suggested by him. 5 References [1] A. 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