HAMILTON DECOMPOSITIONS OF ONE-ENDED CAYLEY GRAPHS

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1 HAMILTON DECOMPOSITIONS OF ONE-ENDED CAYLEY GRAPHS JOSHUA ERDE, FLORIAN LEHNER, AND MAX PITZ Abstract. We prove that any one-ended, ocay finite Cayey graph with non-torsion generators admits a decomposition into edge-disjoint Hamitonian (i.e. spanning) doube-rays. In particuar, the n-dimensiona grid Z n admits a decomposition into n edge-disjoint Hamitonian doube-rays for a n N. 1. Introduction A Hamitonian cyce of a finite graph is a cyce which incudes every verte of the graph. A finite graph G = (V, E) is said to have a Hamiton decomposition if its edge set can be partitioned into disjoint sets E = E 1 E 2 E r such that each E i is a Hamitonian cyce in G. The starting point for the theory of Hamiton decompositions is an od resut by Waecki from 1890 according to which every finite compete graph of odd order has a Hamiton decomposition (see [2] for a description of his construction). Since then, this resut has been etended in various different ways, and we refer the reader to the survey of Aspach, Bermond and Sotteau [3] for more information. Hamitonicity probems have aso been considered for infinite graphs, see for eampe the survey by Gaian and Witte [16]. Whie it is sometimes not obvious which objects shoud be considered the correct generaisations of a Hamitonian cyce in the setting of infinite graphs, for one-ended graphs the undisputed soution is to consider doube-rays, i.e. infinite, connected, 2-reguar subgraphs. Thus, for us a Hamitonian doube-ray is then a doube-ray which incudes every verte of the graph, and we say that an infinite graph G = (V, E) has a Hamiton decomposition if we can partition its edge set into edge-disjoint Hamitonian doube-rays. In this paper we wi consider infinite variants of two ong-standing conjectures on the eistence of Hamiton decompositions for finite graphs. The first conjecture concerns Cayey graphs: Given a finitey generated abeian group (Γ, +) and a finite generating set S of Γ, the Cayey graph G(Γ, S) is the muti-graph with verte set Γ and edge muti-set {(, + g) : Γ, g S}. Conjecture 1 (Aspach [1]). If Γ is an abeian group and S generates G, then the simpification of G(Γ, S) has a Hamiton decomposition, provided that it is 2kreguar for some k Mathematics Subject Cassification. 05C45, 05C63, 20K99. Key words and phrases. Hamiton decomposition; Cayey graph; doube ray; Aspach conjecture. Joshua Erde was supported by the Aeander von Humbodt Foundation. Forian Lehner was supported by the Austrian Science Fund (FWF) Grant no. J 3850-N32. 1

2 2 JOSHUA ERDE, FLORIAN LEHNER, AND MAX PITZ Note that if S S =, then G(Γ, S) is automaticay a 2 S -reguar simpe graph. If G(Γ, S) is finite and 2-reguar, then the conjecture is triviay true. Bermond, Favaron and Maheo [6] showed that the conjecture hods in the case k = 2. Liu [11] proved certain cases of the conjecture for finite 6-reguar Cayey graphs, and his resut was further etended by Westund [15]. Our main theorem in this paper is the foowing affirmative resut towards the corresponding infinite anaogue of Conjecture 1: Theorem 1.1. Let Γ be an infinite, finitey generated abeian group, and et S be a generating set such that every eement of S has infinite order. If the Cayey graph G = G(Γ, S) is one-ended, then it has a Hamiton decomposition. We remark that under the assumption that eements of S are non-torsion, the simpification of G(Γ, S) is aways isomorphic to a Cayey graph G(Γ, S ) with S S and S S =, and so our theorem impies the corresponding version of Conjecture 1 for non-torsion generators, in particuar for Cayey graphs of Z n with arbitrary generators. In the case when G = G(Γ, S) is two-ended, there are additiona technica difficuties when trying to construct a decomposition into Hamitonian doube-rays. In particuar, since each Hamitonian doube-ray must meet every edge cut an odd number of times, there can be parity reasons why no decomposition eists. One particuar two-ended case, namey where Γ = Z, has been considered by Bryant, Herke, Maenhaut and Webb [7], who showed that when G(Z, S) is 4-reguar, then G has a Hamiton decomposition uness there is an odd cut separating the two ends. The second conjecture about Hamitonicity that we consider concerns Cartesian products of graphs: Given two graphs G and H the Cartesian product (or product) G H is the graph with verte set V (G) V (H) in which two vertices (g, h) and (g, h ) are adjacent if and ony if either g = g and h is adjacent to h in H, or h = h and g is adjacent to g in G. Kotzig [10] showed that the Cartesian product of two cyces has a Hamiton decomposition, and conjectured that this shoud be true for the product of three cyces. Bermond etended this conjecture to the foowing: Conjecture 2 (Bermond [5]). If G 1 and G 2 are finite graphs which both have Hamiton decompositions, then so does G 1 G 2. Aspach and Godsi [4] showed that the product of any finite number of cyces has a Hamiton decomposition, and Stong [14] proved certain cases of Conjecture 2 under additiona assumptions on the number of Hamiton cyces in the decomposition of G 1 and G 2 respectivey. Appying techniques we deveoped to prove Theorem 1.1, we show as our second main resut of this paper that Conjecture 2 hods for countaby infinite graphs. Theorem 1.2. If G and H are countabe graphs which both have Hamiton decompositions, then so does their product G H. The paper is structured as foows: In Section 2 we mention some group theoretic resuts and definitions we wi need. In Section 3 we state our main emma, the Covering Lemma, and show that it impies Theorem 1.1. The proof of the Covering Lemma wi be the content of Section 4. In Section 5 we appy our techniques

3 HAMILTON DECOMPOSITIONS OF ONE-ENDED CAYLEY GRAPHS 3 to prove Theorem 1.2. Finay, in Section 6 we ist open probems and possibe directions for further work. 2. Notation and preiminaries If G = (V, E) is a graph, and A, B V, we denote by E(A, B) the set of edges between A and B, i.e. E(A, B) = {(, y) E : A, y B}. For A V or F E we write G[A] and G[F ] for the subgraph of G induced by A and F respectivey. For A, B Γ subsets of an abeian group Γ we write A+B := {a + b: a A, b B} Γ and A := { a: a A}. If is a subgroup of Γ, and A Γ a subset, then A = {a + : a A} denotes the famiy of corresponding cosets. If g Γ we say that the order of g is the smaest k N such that k g = 0. If such a k eists, then g is a torsion eement. Otherwise, we say the order of g is infinite and g is a non-torsion eement. For k N we write [k] = {1, 2,, k}. The foowing terminoogy wi be used throughout. Definition 2.1. Given a graph G, an edge-coouring c: E(G) [s] and a coour i [s], the i-subgraph is the subgraph of G induced by the edge set c 1 (i), and the i-components are the components of the i-subgraph. Definition 2.2 (Standard and amost-standard coourings of Cayey graphs). Let Γ be an infinite abeian group, S = {g 1, g 2,, g s } a finite generating set for Γ such that every g i S has infinite order, and et G be the Cayey graph G(Γ, S). The standard coouring of G is the edge coouring c std : E(G) [s] such that c std ( (, + gi ) ) = i for each Γ, g i S. Given a subset X V (G) we say that a coouring c is standard on X if c agrees with c std on G[X]. Simiary if F E(G) we say that c is standard on F if c agrees with c std on F. A coouring c: E(G) [s] is amost-standard if the foowing are satisfied: there is a finite subset F E(G) such that c is standard on E(G) \ F ; for each i [s] the i-subgraph is spanning, and each i-component is a doube-ray. Definition 2.3 (Standard squares and doube-rays). Let Γ and S be as above. Given Γ and g i g j S, we ca (, g i, g j ) := {(, + g i ), (, + g j ), ( + g i, + g i + g j ), ( + g j, + g i + g j )} an (i, j)-square with base point, and (, g i ) := {( + ng i, + (n + 1)g i ): n Z} an i-doube-ray with base point. Moreover, given a coouring c: E(G(Γ, S)) [s] we ca (, g i, g j ) and (, g i ) an (i, j)-standard square and i-standard doube-ray if c is standard on (, g i, g j ) and (, g i ) respectivey. Since Γ is an abeian group, every (, g i, g j ) is a 4-cyce in G(Γ, S) (provided g i g j ), and since S contains no torsion eements of Γ, (, g k ) reay is a doube-ray in the Cayey graph G(Γ, S). Let Γ be a finitey generated abeian group. By the Cassification Theorem for finitey generated abeian groups (see e.g. [9]), there are integers n, q 1,, q r such that Γ = Z n r i=1 Z q i, where Z q is the additive group of the integers moduo q.

4 4 JOSHUA ERDE, FLORIAN LEHNER, AND MAX PITZ In particuar, for each Γ there is an integer n and a finite abeian group Γ fin such that Γ = Z n Γ fin. The foowing structura theorem for the ends of finitey generated abeian groups is we-known: Theorem 2.4. For a finitey generated group Γ = Z n Γ fin, the foowing are equivaent: n 2, there eists a finite generating set S such that G(Γ, S) is one-ended, and for a finite generating sets S, the Cayey graph G(Γ, S) is one-ended. Proof. See e.g. [13, Proposition 5.2] for the fact the number of ends of G(Γ, S) is independent of the choice of the generating set S, and [13, Theorem 5.12] for the equivaence with the first item. A group Γ satisfying one of the conditions from Theorem 2.4 is caed one-ended. Coroary 2.5. Let Γ be an abeian group, S = {g 1,, g s } be a finite generating set such that the Cayey graph G(Γ, S) is one-ended. Then, for every g i S of infinite order, there is some g j S such that g i, g j = (Z 2, +). Proof. Suppose not. It foows that in Γ/ g i every eement has finite order, and since it is aso finitey generated, it is some finite group Γ f such that Γ = Z Γ f. Thus, by Theorem 2.4, G is not one-ended, a contradiction. 3. The covering emma and a high-eve proof of Theorem 1.1 Every Cayey graph G(Γ, S) comes with a natura edge coouring c std, where we coour an edge (, + g i ) with Γ and g i S according to the inde i of the corresponding generating eement g i. If every eement of S has infinite order, then every i-subgraph of G(Γ, S) consists of a spanning coection of edge-disjoint doube-rays, see Definitions 2.1 and 2.2. So, it is perhaps a natura strategy to try to buid a Hamitonian decomposition by combining each of these monochromatic coections of doube-rays into a singe monochromatic spanning doube-ray. Rather than trying to do this directy, we sha do it in a series of steps: given any coour i [s] = S and any finite set X V (G), we wi show that one can change the standard coouring at finitey many edges so that there is one particuar doube-ray in the coour i which covers X. Moreover, we can ensure that the resuting coouring maintains enough of the structure of the standard coouring that we can repeat this process inductivey: it shoud remain amost standard, i.e. a monochromatic components are sti doube-rays, see Definition 2.2. By taking a sequence of sets X 1 X 2 ehausting the verte set of G, and varying which coour i we consider, we ensure that in the imit, each coour cass consists of a singe spanning doube-ray, giving us the desired Hamiton decomposition. In this section, we formuate our key emma, namey the Covering Lemma 3.1, which aows us to do each of these steps. We wi then show how Theorem 1.1 foows from the Covering Lemma. The proof of the Covering Lemma is given in Section 4. Lemma 3.1 (Covering emma). Let Γ be an infinite, one-ended abeian group, S = {g 1, g 2,, g s } a finite generating set such that every g i S has infinite order, and G = G(Γ, S) the corresponding Cayey graph.

5 HAMILTON DECOMPOSITIONS OF ONE-ENDED CAYLEY GRAPHS 5 Then for every amost-standard coouring c of G, every coour i and every finite subset X V (G), there eists an amost-standard coouring ĉ of G such that ĉ = c on E(G[X]), and some i-component in ĉ covers X. Proof of Theorem 1.1 given Lemma 3.1. Fi an enumeration V (G) = {v n : n N}. Let X 0 = D 0 = {v 0 } and c 0 = c std. For each n 1 we wi recursivey construct amost standard coourings c n : E(G) [s], finite subsets X n V (G), (n mod s)-components D n of c n and finite paths D n D n such that for every n N (1) X n 1 {v n } X n, (2) V (D n 1) X n, (3) X n V (D n), (4) D n propery etends the path D n s (the previous path of coour n mod s) in both endpoints of D n s, and (5) c n agrees with c n 1 on E(G[X n ]). Suppose inductivey for some n N that c n, X n, D n and D n have aready been defined. Choose some X n+1 X n {v n } arge enough such that (1) and (2) are satisfied. Appying Lemma 3.1 with input c n and X n+1 provides us with a coouring c n+1 such that (5) is satisfied and some (n + 1 mod s)-component D n+1 covers X n+1. Since c n+1 is amost standard, D n+1 is a doube-ray. Furthermore, since c n+1 agrees with c n on E(G[X n+1 ]), by the inductive hypothesis it agrees with c k on E(G[X k+1 ]) for each k n. Therefore, since D n+1 s X n s+2 is a path of coour (n + 1 mod s) in c n+1 s, it foows that D n+1 s D n+1 and so we can etend D n+1 s to a sufficienty ong finite path D n+1 D n+1 such that (3) and (4) are satisfied at stage n + 1. Once the construction is compete, we define T 1,, T s G by T i = n i mod s and caim that they form a decomposition of G into edge-disjoint Hamitonian doube-rays. Indeed, by (4), each T i is a doube-ray. That they are edge-disjoint can be seen as foows: Suppose for a contradiction that e E(T i ) E(T j ). Choose n(i) and n(j) minima such that e E(D n(i) ) E(T i) and e E(D n(j) ) E(T j). We may assume that n(i) < n(j), and so e E(G[X n(i)+1 ]) by (2). Furthermore, by (5) it foows that c n(j) agrees with c n(i) on E(G[X n(i)+1 ]). However by construction c n(j) (e) = j i = c n(i) (e) contradicting the previous ine. Finay, to see that each T i is spanning, consider some v n V (G). By (1), v n X n. Pick n n with n i mod s. Then by (3), D n T i covers X n which in turn contains v n, as v n X n X n by (1). D n 4. Proof of the Covering Lemma 4.1. Banket assumption. Throughout this section, et us now fi a one-ended infinite abeian group Γ with finite generating set S = {g 1,, g s } such that every eement of S has infinite order, an amost-standard coouring c of the Cayey graph G = G(Γ, S), a finite subset X Γ such that c is standard on V (G) \ X, a coour i, say i = 1, and corresponding generator g 1 S, for which we want to show Lemma 3.1, and finay

6 6 JOSHUA ERDE, FLORIAN LEHNER, AND MAX PITZ a second generator in S, say g 2, such that := g 1, g 2 = (Z 2, +), see Coroary Overview of proof. We want to show Lemma 3.1 for the Cayey graph G, coouring c, generator g 1 and finite set X. The cosets of g 1, g 2 in Γ cover V (G), and in the standard coouring the edges of coour 1 and 2 form a grid on g 1, g 2. So, since c is amost-standard, on each of these cosets the edges of coour 1 and 2 wi ook ike a grid, apart from on some finite set. Our aim is to use the structure in these grids to change the coouring c to one satisfying the concusions of Lemma 3.1. It wi be more convenient to work with arge finite grids, which we require, for technica reasons, to have an even number of rows. This is the reason for the sight asymmetry in the definition beow. Notation 4.1. Let g i, g j Γ. For N, M N we write g i, g j N,M := {ng i + mg j : n, m Z, N n N, M < m M} g i, g j Γ. The structure of our proof can be summarised as foows. First, in Section 4.3, we wi show that there is some N 0 and some nice finite set of P of representatives of cosets of g 1, g 2 such that P + g 1, g 2 N0,N 0 covers X. We wi then, in Section 4.4 pick sufficienty arge numbers N 0 < N 1 < N 2 < N 3 and consider the grids P + g 1, g 2 N3,N 1. Using the structure of the grids we wi make oca changes to the coouring inside P + ( g 1, g 2 N3,N 1 ) to construct our new coouring ĉ. This new coouring ĉ wi then agree with c on the subgraph induced by P + g 1, g 2 N0,N 0 X, and be standard on V (G) \ ( ) P + g 1, g 2 N3,N 1, and hence, as ong as we ensure a the coour components are doube-rays, amost-standard. These oca changes wi happen in three steps. First, in Step 1, we wi make oca changes inside + ( g 1, g 2 N3,N 1 \ g 1, g 2 N2,N 1 ) for each P, in order to make every i-component meeting P + g 1, g 2 N2,N 1 a finite cyce. Net, in Step 2, we wi make oca changes inside +( g 1, g 2 N2,N 1 \ g 1, g 2 N1,N 1 ) for each P, in order to combine the cyces meeting this transate of the grid into a singe cyce. Finay, in Step 3, we wi make oca changes inside P +( g 1, g 2 N1,N 1 \ g 1, g 2 N0,N 0 ), in order to join the cyces for different into a singe cyce covering P + g 1, g 2 N0,N 0. We then make one fina oca change to turn this finite cyce into a doube-ray Identifying the reevant cosets. Lemma 4.2. There eist N 0 N and a finite set P = { 0,, t } Γ such that P = { 0 +,, t + } is a path in G(Γ/, (S \ {g 1, g 2 }) ), and X P + g 1, g 2 N0,N 0. Proof. Since X is finite, there is a finite set Y = {y 1,, y k } Γ such that the cosets in Y = {y 1 +,, y k + } are a distinct and cover X. Moreover, since every (y + ) X is finite, there eists N 0 N such that (y + g 1, g 2 ) X = (y + g 1, g 2 N0,N 0 ) X for a 1 k. Then X Y + g 1, g 2 N0,N 0. Net, by a resut of Nash-Wiiams [12], every Cayey graph of a countaby infinite abeian group has a Hamiton doube-ray, and it is a fokore resut (see [16]) that every Cayey graph of a finite abeian group has a Hamiton cyce. So in particuar, the Cayey graph of (Γ/, (S \ {g 1, g 2 }) ), has a Hamiton cyce /

7 HAMILTON DECOMPOSITIONS OF ONE-ENDED CAYLEY GRAPHS 7 doube-ray, say H. Let P Y be a finite set of representatives of the cosets of which ie on the conve hu of Y on H. It is cear that P is as required. For the rest of this section et us fi N 0 N and P = { 0,, t } Γ to be as given by Lemma Picking sufficienty arge grids. In order to choose our grids arge enough to be abe to make a the necessary changes to our coouring, we wi first need the foowing emma, which guarantees that we can find, for each k 1, 2 and Γ, many distinct standard k-doube-rays which go between the cosets + and ( + g k ) +. Lemma 4.3. For any g k S \ {g 1, g 2 } and any pair of distinct cosets + and (g k + ) +, there are infinitey many distinct standard k-doube-rays R for the coouring c with E(R) E( +, (g k + ) + ). Proof. It ceary suffices to prove the assertion for c = c std. We caim that either R 1 = { ( + mg 1, g k ): m Z} or R 2 = { ( + mg 2, g k ): m Z} is such a coection of disjoint standard k-doube-rays. Suppose that R 1 is not a coection of disjoint doube-rays. Then there are m m Z and n, n Z such that mg 1 + ng k = m g 1 + n g k. Since g k has infinite order, it foows that n n, too, and so we can concude that there are, Z \ {0} such that g 1 = g k. Simiary, if R 2 was not a coection of disjoint doube-rays, then we can find q, q Z \ {0} such that qg 2 = q g k. However, it now foows that q g 1 = q ( g k ) = (q g k ) = qg 2, contradicting the fact that g 1, g 2 = (Z 2, +). This estabishes the caim. Finay, observe that if say R 1 is a disjoint coection, then for every R m = ( + mg 1, g k ) R 1 we have ( + mg 1, + mg 1 + g k ) E(R m ) E( +, (g k + ) + ) as desired. We are now ready to define our numbers N 0 < N 1 < N 2 < N 3. Reca that N 0 and P = { 0,, t } are given by Lemma 4.2. For each [t], et g n() be some generator in S \ {g 1, g 2 } that induces the edge between 1 + and + on the path P. Note that n() [s] \ {1, 2} for a. By Lemma 4.3, we may find t 2 many disjoint standard doube-rays R = { R k : 1 k, t } such that for every, the doube-rays in { R k = ( y k, g n()) : k [t] } are standard n()-doube-rays containing an edge e k = (y k, y k + g n() ) E(R k ) E( 1 +, + ) so that a T k = ( y k, g ) i, g n() are (1, n())-standard squares for c which have empty intersection with { 1, } + g 1, g 2 N0,N 0. Furthermore we may assume that these standard squares are a edge-disjoint. Then et N 1 > N 0 be sufficienty arge such that the subgraph induced by P + g 1, g 2 N1 3,N 1 3 contains a standard squares T k mentioned above. Let N 2 be arbitrary with N 2 5N 1. Let N 3 be arbitrary with N 3 N 2 + 2N 1.

8 8 JOSHUA ERDE, FLORIAN LEHNER, AND MAX PITZ 4.5. The cap-off step. Our main too for ocay modifying our coouring is the foowing notion of coour switchings, which is aso used in [11]. Definition 4.4 (Coour switching of standard squares). Given an edge coouring c: E(G(Γ, S)) [s] and an (i, j)-standard square (, g i, g j ), a coour switching on (, g i, g j ) changes the coouring c to the coouring c such that c = c on E \ (, g i, g j ), c ( (, + g i ) ) = c ( ( + g j, + g i + g j ) ) = j, c ( (, + g j ) ) = c ( ( + g i, + g i + g j ) ) = i. It woud be convenient if coour switchings maintained the property that a coouring is amost-standard. Indeed, if c is standard on E(G) \ F then c is standard on E(G) \ (F (, g i, g j )). Aso, it is a simpe check that if the i and j-subgraphs of G for c are 2-reguar and spanning, then the same is true for c. However, some i or j-components may change from doube-rays to finite cyces, and vice versa. Step 1 (Cap-off step). There is a coouring c obtained from c by coour switchings of finitey many (1, 2)-standard squares such that c = c on E(G[X]); every 1-component in c meeting P + g 1, g 2 N2,N 1 is a finite cyce intersecting both P + ( g 1, g 2 N3,N 1 \ g 1, g 2 N2,N 1 ) and P + g 1, g 2 N1,N 1 ; every other 1-component, and a other components of a other coour casses of c are doube-rays; c is standard outside of P + g 1, g 2 N3,N 1 and inside of P + ( g 1, g 2 N2,N 1 \ g 1, g 2 N0,N 0 ); for each P, the sets of vertices { + ng 1 + mg 2 : N 1 n N 2, m {N 1, N 1 1}} are each contained in a singe 1-component of c. Proof. For [t] and q [N 1 ] et R q = ( v q, g 1, g 2 ) and L q = ( w q, g 1, g 2 ) be the (1, 2)-squares with base point v q = + (N q) g 1 + (N q) g 2 and w q = (N q) g 1 + (N q) g 2 respectivey. The square L q is the mirror image of R q with respect to the y-ais of the grid + g 1, g 2, however the base points are not mirror images, accounting for the sight asymmetry in the definitions. Since N 3 N 2 + 2N 1, it foows that R q L q E( + ( g 1, g 2 N3,N 1 \ g 1, g 2 N2,N 1 )) for a q [N 1 ], and so by assumption on c, a R q and L q are indeed standard (1, 2)- squares. We perform coour switchings on R q and L q for a [t] and q [N 1 ], and ca the resuting edge coouring c. It is cear that c = c on E(G[X]) and that c is standard outside of P + g 1, g 2 N3,N 1 and inside of P +( g 1, g 2 N2,N 1 \ g 1, g 2 N0,N 0 ). Let C G denote the region consisting of a vertices that ie in + ( g 1, g 2 N3,N 1 for some between a pair L q and R q for some q, i.e. C = t N 1 =1 q=1 { + ng 1 + (N q)g 2, + ng 1 + (N q)g 2 : n N q}. Then P + g 1, g 2 N2,N 1 C. By construction, there are no edges of coour 1 in c eaving C, that is, E(C, V (G)\C) c 1 (1) =. In particuar, since the 1-subgraph

9 HAMILTON DECOMPOSITIONS OF ONE-ENDED CAYLEY GRAPHS 9 + g 1, g 2 N0,N 0 g 1, g 2 N1,N 1 g 1, g 2 N2,N 1 g 1, g 2 N3,N1 Figure 1. Performing coour switchings of standard squares at positions indicated by in a copy + g 1, g 2 N3,N 1 of a finite grid. of G under c remains 2-reguar and spanning, as remarked above, a 1-components under c inside C are finite cyces, whose union covers C. Aso, since each 1-component of c is a doube-ray, it must eave the finite set P + g 1, g 2 N3,N 1 and hence meets some Rq or L q. Therefore, by construction each 1-component of c inside C meets some Rq or L q and so, since c is standard outside of P + g 1, g 2 N0,N 0 ecept at the squares Rq or L q, each such 1-component meets both P + ( g 1, g 2 N3,N 1 \ g 1, g 2 N2,N 1 ) and P + g 1, g 2 N1,N 1. Moreover, a other coour components remain doube-rays. This is cear for a k- components of G if k 1, 2 (as the coours switchings of (1, 2)-standard squares did not affect these other coours). However, it is aso cear for the 1-cooured douberays outside of C and aso for a 2-cooured components, as we chose our standard squares Rq and L q staggered, so as not to create any finite monochromatic cyces, see Figure 1 (reca that every + is isomorphic to the grid). Finay, since N 1 > N 0, the edge set {( + ng 1 + N 1 g 2, + (n + 1)g 1 + N 1 g 2 ): N 3 n < N 3 1} { (v 1, v 1 + g 2 ), ((w 1 + g 1, w 1 + g 1 + g 2 )) } {( + ng 1 + (N 1 1)g 2, + (n + 1)g 1 + (N 1 1)g 2 )} : N 3 n < N 1 {( + ng 1 + (N 1 1)g 2, + (n + 1)g 1 + (N 1 1)g 2 )} : N 1 n < N 3 meets ony R1 and L 1 and therefore is easiy seen to be part of the same 1- component of c. In Figure 1, these edges correspond to the red ine at the top, and the two ines beow it on either side of + g 1, g 2 N1,N Combining cyces inside each coset of. In the previous step we chose the (1, 2)-standard squares at which we performed coour switchings in a staggered manner in the grids + g 1, g 2 N3,N 1, so that we coud guarantee that a the 2- components were sti doube-rays afterwards. In ater steps we wi no onger be abe to be as epicit about which standard squares we perform coour switchings at, and so we wi require the foowing definitions to be abe to say when it is safe to perform a coour switching at a standard square.

10 10 JOSHUA ERDE, FLORIAN LEHNER, AND MAX PITZ Definition 4.5 (Crossing edges). Suppose R = {(v i, v i+1 ): i Z} is a doube-ray and e 1 = (v j1, v j2 ) and e 2 = (v k1, v k2 ) are edges with j 1 < j 2 and k 1 < k 2. We say that e 1 and e 2 cross on R if either j 1 < k 1 < j 2 < k 2 or k 1 < j 1 < k 2 < j 2. Lemma 4.6. For an edge-coouring c: E(G(Γ, S)) [s], suppose that (, g i, g k ) is an (i, k)-standard square with g i g k, and further that the two k-cooured edges (, + g k ) and ( + g i, + g i + g k ) of (, g i, g k ) ie on the same standard k- doube-ray R = (, g k ). Then the two i-cooured edges of (, g i, g k ) cross on R. Proof. Write e 1 = (, + g i ) and e 2 = ( + g k, + g k + g i ) for the two i-cooured edges of (, g i, g k ). The assumption that (, + g k ) and ( + g i, + g i + g k ) both ie on (, g k ) impies that g i = rg k for some r Z \ { 1, 0, 1}. If r > 1, we have < + g k < + g i < + g k + g i (where < denotes the natura inear order on the verte set of the doube-ray), and if r < 1, we have +g i < +g k +g i < < +g k, and so the edges e 1 and e 2 indeed cross on R. Definition 4.7 (Safe standard square). Given an edge coouring c: E(G(Γ, S)) [s] we say an (i, k)-standard square (, g i, g k ) is safe if g i g k and either the k-components for c meeting T are distinct doube-rays, or there is a unique k-component for c meeting T, which is a doube-ray on which (, + g i ) and ( + g k, + g i + g k ) cross. The foowing emma tes us, amongst other things, that if we perform a coour switching at a safe (1, k)-standard square then the k-components in the resuting coouring meeting that square wi sti be doube-rays. Lemma 4.8. Let c: E(G(Γ, S)) [s] be an edge coouring, T = (, g i, g k ) be an (i, k)-standard square with g i g k, and c be the coouring obtained by performing a coour switching on T. Suppose that the i and k-components for c meeting T are a 2-reguar, and that there are two distinct i-components C 1 and C 2 meeting T, at east one of which is a finite cyce. Then the foowing statements are true: There is a singe i-component for c meeting T which covers V (C 1 ) V (C 2 ); If the k-components for c meeting T are distinct doube-rays then the k- components for c meeting T are distinct doube-rays; If there is a unique k-component for c meeting T, which is a doube-ray on which (, + g i ) and ( + g k, + g i + g k ) cross, then there is unique k-component for c meeting T, which is a doube-ray. Proof. Let us write e i = (, + g i ), e k = (, + g k ), e i = ( + g k, + g i + g k ) and e k = ( + g i, + g i + g k ), so that (, g i, g j ) = {e i, e k, e i, e k }. For the first item, et the i-components for c be e i C 1 and e i C 2, where without oss of generaity C 2 is a finite cyce. Then C 2 e i is a finite path, and C 1 e i has at most 2 components, one containing and one containing +g i. Hence, the i-component for c meeting T, (C 1 C 2 ) {e i, e i } + {e k, e k }, is connected and covers V (C 1 ) V (C 2 ). For the second item, et the k-components for c be e k D 1 and e k D 2. Then D 1 e k has two components, a ray starting at and a ray starting at + g k. Simiary, D 2 e k has two components, a ray starting at + g i and a ray starting at +g i +g k. Hence, the k-components for c meeting T, which are the components of (D 1 D 2 ) {e k, e k } + {e i, e i }, are distinct doube-rays.

11 HAMILTON DECOMPOSITIONS OF ONE-ENDED CAYLEY GRAPHS 11 e k e i e i e k e i e k e i e k Figure 2. The two situations in Lemma 4.8 with i in red and k in bue. Finay, if there is a singe k-component D for c meeting T such that D is a doube-ray, then D {e k, e k } consist of three components. Since e i and e i cross on D there are two cases as to what these components are. Either the components consist of two rays, starting at and + g i + g k and a finite path from + g k to + g i, or the components consist of two rays, starting at + g i and + g k, and a finite path from + g i + g k to. In either case, the k-component for c meeting T, namey D {e k, e k } + {e i, e i }, is a doube-ray. Lemma 4.8 is aso usefu as the first item aows us to use (1, k) coour switchings to combine two 1-components into a singe 1-component which covers the same verte set. Step 2 (Combining cyces step). We can change c from Step 1 via coour switchings of finitey many (1, 2)-standard squares to a coouring c satisfying c = c = c on E(G[X]); every 1-component in c meeting P + g 1, g 2 N2,N 1 is a finite cyce intersecting both P + ( g 1, g 2 N3,N 1 \ g 1, g 2 N2,N 1 ) and P + g 1, g 2 N1,N 1 ; every other 1-component, and a other components of a other coour casses of c are doube-rays; every 1-component in c meeting some k + ( g 1, g 2 N2,N 1 ) covers k + ( g 1, g 2 N2,N 1 ); c is standard outside of P + g 1, g 2 N3,N 1 and inside of P + ( g 1, g 2 N1,N 1 \ g 1, g 2 N0,N 0 ). Proof. Our pan wi be to go through the grids k + g 1, g 2 N2,N 1 in order, from k = 0 to t, and use coour switchings to combine a the 1-components which meet k + ( g 1, g 2 N2,N 1 ) into a singe 1-component. We note that, since c is not standard on X, it may be the case that these 1-components aso meet k + g 1, g 2 N2,N 1 for k k. We caim inductivey that there eists a sequence of coourings c = c 0, c 1,, c t = c such that for each 0 t: c = c = c on E(G[X]); every 1-component in c meeting P + g 1, g 2 N2,N 1 is a finite cyce intersecting both P + ( g 1, g 2 N3,N 1 \ g 1, g 2 N2,N 1 ) and P + g 1, g 2 N1,N 1 ; for every k, every 1-component in c meeting k + ( g 1, g 2 N2,N 1 \ g 1, g 2 N0,N 0 ) covers k + ( g 1, g 2 N2,N 1 ); for every k >, c = c on k + g 1, g 2 N2,N 1

12 12 JOSHUA ERDE, FLORIAN LEHNER, AND MAX PITZ every other 1-component, and a other components of a other coour casses of c are doube-rays; c is standard outside of P + g 1, g 2 N3,N 1 and inside of P + ( g 1, g 2 N1,N 1 \ g 1, g 2 N0,N 0 ). In Step 1 we constructed c 0 = c such that this hods. Suppose that 0 < t, and that we have aready constructed c k for k <. For q [4N 1 2] we define T q = (v q, g 1, g 2 ) to be the (1, 2)-square with base point { + (N q)g 1 + (N 1 q)g 2 if q 2N 1 1, and v q = (N q )g 1 + (N 1 q )g 2 if q = q (2N 1 1) 1. With these definitions, T 2N1 1+q is the mirror image of T q for a q [2N 1 1] aong the y-ais. Moreover, since N 2 5N 1, each T q is contained within k + ( g 1, g 2 N2,N 1 \ g 1, g 2 N1,N 1 ). We wi combine the 1-components in c 1 which meet + ( g 1, g 2 N2,N 1 \ g 1, g 2 N0,N 0 ) into a singe component by performing coour switchings at some of the (1, 2)-squares T q. Let us show first that most of the induction hypotheses are maintained regardess of the subset of the T q we make switchings at. + g 1, g 1 N0,N 0 g 1, g 2 N1,N 1 g 1, g 2 N2,N1 Figure 3. The standard squares T q, with a coour switching performed at T 2. We note that, since c 1 is standard inside of + ( g 1, g 1 N2,N 1 ) and outside of P + g 1, g 2 N3,N 1, and g 1 g 2, each T q is a safe (1, 2)-standard square for c 1. Furthermore, by construction, even if we perform coour switchings at any subset of the T q, the remaining squares remain standard and safe. Hence, by Lemma 4.8 and the induction assumption, after performing coour switchings at any subset of the standard squares T q a 2-components of the resuting coouring wi be doube-rays. Secondy, these coour switching wi not change the coouring outside of P + g 1, g 2 N2,N 1 and inside of P + g 1, g 2 N1,N 1, or in any k + g 1, g 2 N2,N 1 with k. In particuar, every 1-component not meeting P + g 1, g 2 N2,N 1 wi sti be a doube-ray. Finay, again by Lemma 4.8, every 1-component of the resuting coouring meeting P + g 1, g 2 N2,N 1 wi be a finite cyce which covers the verte set of some union of 1-components in c 1, and hence wi intersect both P + ( g 1, g 2 N3,N 1 \ g 1, g 2 N2,N 1 ) and P + g 1, g 2 N1,N 1. Let us write e q = (v q, v q + g 1 ) for each q [4N 1 2]. Since c 1 = c on + g 1, g 2 N2,N 1, and by Step 1 c is standard on +( g 1, g 2 N2,N 1 \ g 1, g 2 N0,N 0 ), each 1-component of c 1 that meets + ( g 1, g 2 N2,N 1 ) contains at

13 HAMILTON DECOMPOSITIONS OF ONE-ENDED CAYLEY GRAPHS 13 east one e q. Aso, e 1 and e 2N1 beong to the same 1-component by the ast caim in Step 1. Let us write C for the coection of such cyces, and consider the map α: C {1,, 4N 1 1}, C min {q : e q E(C)}, which maps each cyce to the first e q that it contains. Since C is a disjoint coection of cyces, the map α is injective. Now et c be the coouring obtained from c 1 by switching a standard squares in T = {T q : q ran(α)} \ {T 1 }. We caim that c satisfies our induction hypothesis for. By the previous comments it wi be sufficient to show Caim 1. Every 1-component in c meeting +( g 1, g 2 N2,N 1 \ g 1, g 2 N0,N 0 ) covers + ( g 1, g 2 N2,N 1 ). To see this, we inde C = {C 1,, C r } such that u < v impies α(c u ) < α(c v ), and consider the sequence of coourings {c z : z [r]} where c 1 = c and each c z is obtained from c z 1 by switching the standard square T α(cz). Let us show by induction that for every z [r] there is an 1-component of c z which covers y z C y. For z = 1 the caim is ceary true. So, suppose z > 1. Since α(c z ) is minima in {α(c y ): y z} it foows that e q y<z C y for every q < α(c z ). Note that, since c 1 = c on + g 1, g 2 N2,N 1, it foows from the fina caim in the Cap-off step that C 1 contains both e 1 and e 2N1, and so α(c z ) 2N 1. Consider the standard square T α(cz). Since c 1 = c on + g 1, g 2 N2,N 1, by construction the edge opposite to e α(cz) in T α(cz), that is, e α(cz) + g j, is in the same 1-component in c 1 as e α(cz) 1, and hence is contained in y<z C y. Therefore, by Lemma 4.8, after performing an (1, 2)-coour switching at T α(cz), the 1-component in c z contains y z C y. Hence, there is an 1-component of c = c r which covers y r C y, and so there is a unique 1-component of c meeting + ( g 1, g 2 N2,N 1 ) which covers it, estabishing the caim Combining cyces across different cosets of. In the third and fina step we join the finite cyces covering each + ( g 1, g 2 N1,N 1 ) into a singe finite cyce, and then make one fina switch to absorb this cyce into a doube-ray. The resuting coouring wi then satisfy the conditions of Lemma 3.1. Step 3 (Combining cosets step). We can change c from the previous emma to an amost-standard coouring ĉ such that ĉ = c = c = c on E(G[X]); Some component in coour 1 covers P + g 1, g 2 N1,N 1. Proof. Reca that P = { 0,, t } is such that P = { 0 +,, t + } is a finite, graph-theoretic path in the Cayey graph of the quotient Γ/ with generating set S \ {g 1, g 2 }. Moreover, reca from Section 4.4 that N 1 > N 0 was chosen so that for the initia coouring c there were t 2 many disjoint standard doube-rays R = { R k : 1 k, t } such that for every, the doube-rays in { R k = ( y k, g n()) : k [t] } are standard n()-doube-rays containing an edge e k = (y k, y k + g n() ) E(R k ) E( 1 +, + )

14 14 JOSHUA ERDE, FLORIAN LEHNER, AND MAX PITZ so that a T k = ( y k, g ) 1, g n() are edge-disjoint (1, n())-standard squares for the coouring c contained in the subgraph induced by P + g 1, g 2 N1 3,N 1 3 which have empty intersection with { 1, } + g 1, g 2 N0,N 0. However, since we ony atered the (1, 2)-subgraphs of G in Step 1 and 2, it is cear that a these standard douberays and standard squares for c remain standard aso for the coourings c and in particuar c. g n(1) gn(2) g 1, g 2 N0,N g 1, g 2 N0,N g 1, g 2 N0,N g 1, g 2 N1,N g 1, g 2 N1,N g 1, g 2 N1,N 1 Figure 4. Using (1, n())-standard squares to join up different cosets. For this picture, we assume wog that +1 = + g n(+1). We caim that there eists a function k : [t] [t] { } such that iterativey switching T k() (or not doing anything at a if k() = ) resuts in a sequence of coourings c = c 0, c 1,, c t such that for each 0 t, (1) a singe finite 1-component in c covers { 0,, }+( g 1, g 2 N1,N 1 \ g 1, g 2 N0,N 0 ), (2) for every k, every 1-component in c meeting k +( g 1, g 2 N1,N 1 \ g 1, g 2 N0,N 0 ) is a finite cyce covering k + ( g 1, g 2 N1,N 1 ), and (3) every other 1-component, and a other components of a other coour casses in c are doube-rays. In Step 2 we constructed a coouring c 0 = c for which properties (1) (3) are satisfied. Now suppose{ that 1, and that } the coouring c 1 obtained by switching the standard squares T k( ) : [ 1] satisfies (1) (3). By construction, each such standard square T k( ) is incident with the ray R k( ) and potentiay one further n( )-component. But since we had reserved more that 1 different rays R 1,, Rt, it foows that some ray Rk() remains a standard n()-cooured component for c 1. Both edges (y k(), y k() + g i ) and (y k() + g n(), y k() + g n() + g i ) of T k() are contained in { 1, }+( g 1, g 2 N1,N 1 \ g 1, g 2 N0,N 0 ), and hence are, by assumption (2), covered by finite 1-cyces in c 1. If both edges ie in the same finite 1-cyce,

15 HAMILTON DECOMPOSITIONS OF ONE-ENDED CAYLEY GRAPHS 15 there is nothing to do (and we redefine k() :=, and et c = c 1 ). However, if they ie on different finite cyces, we perform a coour switching on the standard square T k() and caim that the resuting c is as required. By Lemma 4.8, the two finite 1-components merge into a singe finite cyce, and so (1) and (2) are certainy satisfied for c. To see (3), we need to verify that T k() is, when we perform the switching, safe. However, T k() was chosen so that the edge (y k(), y k() + g n() ) T k() ies on a standard doube-ray R = R k() of c 1. Aso, by the inductive assumption (3), the second n()-cooured edge (y k() + g i, y k() + g i + g n() ) T k() ies on an n()- cooured doube-ray R in c 1. If R and R are distinct, then T k() is safe, and if R = R then, since R is a standard n()-doube-ray, Lemma 4.6 impies that T k() is safe. Hence c satisfies (3). This competes the induction step. Thus, by (1) and (3), we obtain an edge-coouring c t for G such that a singe finite 1-component covers P + ( g 1, g 2 N1,N 1 ), and a other 1-components and a other components of other coour casses in c t are doube-rays. Furthermore, since every 1-component which meets P + g 1, g 2 N0,N 0 must meet P +( g 1, g 2 N1,N 1 \ g 1, g 2 N0,N 0 ), it foows that the 1-component in fact covers P + g 1, g 2 N0,N 0. Moreover, since T k() P + g 1, g 2 N1 3,N 1 3 for a [t], it foows that c t is standard on 0 + ( g 1, g 2 N1, \ g 1, g 2 N1 3,N 1 3), and that it is standard outside of P + g 1, g 2 N3,N 1. Hence, the square (, g 1, g 2 ) with base point = 0 + (N 1 2)g 1 + N 1 g 2 is a standard (1, 2)-square such that the edge (, + g 1 ) ies on the finite 1-cyce of c t, the edge ( + g 2, + g 2 + g 1 ) ies on standard 1-doube-ray ( + g 2, g 1 ) (ying competey outside of P + g 1, g 2 N3,N 1 ) of c t, and the edges (, +g 2 ) and (+g 1, +g 2 +g 1 ) ie on distinct standard 2-douberays (, g 2 ) and ( + g 1, g 2 ) 0 +( g 1, g 2 N1, \ g 1, g 2 N1 3,N 1 3). Therefore, we may perform a coour switching on (, g 1, g 2 ), which resuts, by Lemma 4.8, in an amost standard coouring of G such that a singe 1-component covers P + g 1, g 2 N1,N 1, and hence X. 5. Hamitonian decompositions of products The techniques from the previous section can aso be appied to give us the foowing genera resut about Hamitonian decompositions of products of graphs. Theorem 1.2. If G and H are countabe graphs which both have Hamiton decompositions, then so does their product G H. Proof. Suppose that {R i : i I} and {S j : j J} form decompositions of G and H into edge-disjoint Hamitonian doube-rays, where I, J may be finite or countaby infinite. Note that, for each i I, j J, R i S j is a spanning subgraph of G H, and is isomorphic to the Cayey graph of (Z 2, +) with the standard generating set. Let π G : G H G and π H : G H H the projection maps from G H onto the respective coordinates. As our standard coouring for G H we take the map { i if e π c: E(G H) I J, 1 G e (E(R i)), j if e π 1 H (E(S j)). Then each R i S j is 2-cooured (with coours i and j), and this coouring agrees with the standard coouring of C Z 2 = G((Z 2, +), {(1, 0), (0, 1)}) from Section 3.

16 16 JOSHUA ERDE, FLORIAN LEHNER, AND MAX PITZ We may suppose that V (G) = N = V (H). Fi a surjection f : N I J such that every coour appears infinitey often. By starting with c 0 = c and appying Lemma 3.1 recursivey inside the spanning subgraphs R f(k) S 1, if f(k) I, or inside R 1 S f(k), for f(k) J, we find a sequence of edge-coourings c k : G H I J and natura numbers M k N k < M k+1 such that c k+1 agrees with c k on the subgraph of G H induced by [0, M k+1 ] 2, there is a finite path D k of coour f(k) in c k covering [0, N k ] 2, and M k+1 is arge enough such that D k [0, M k+1 ] 2. To be precise, suppose we aready have a finite path D k of coour f(k) in c k covering [0, N k ] 2, and at stage k + 1 we have say f(k + 1) I, and so we are considering R f(k+1) S 1 = CZ 2. We choose M k+1 > N k arge enough such that D k [0, M k+1 ] 2 G H, and N k+1 > M k+1 arge enough such that Q 1 = [0, N k+1 ] 2 G H contains a edges where c k differs from the standard coouring c. Net, consider an isomorphism h: R f(k+1) S 1 = CZ 2. Pick a square Q 2 R f(k+1) S 1 with Q 1 Q 2, i.e. a set Q 2 such that h restricted to Q 2 is an isomorphism to the subgraph of C Z 2 induced by [ Ñk+1, Ñk+1] 2 Z 2 for some Ñk+1 N, and then appy Lemma 3.1 to R f(k+1) S 1 and Q 2 to obtain a finite path D k+1 of coour f(k + 1) in c k+1 covering Q 2. It foows that the doube-rays {T i : i I} {T j : j J} with T = k f 1 () D k give the desired decomposition of G H. 6. Open Probems As mentioned in Section 2, the finitey generated abeian groups can be cassified as the groups Z n r i=1 Z q i, where n, r, q 1,, q r Z. Theorem 1.1 shows that Aspach s conjecture hods for every such group with n 2, as ong as each generator has infinite order. The question however remains as to what can be said about Cayey graphs G(Γ, S) when S contains eements of finite order. Probem 1. Let Γ be an infinite, finitey-generated, one-ended abeian group and S be a generating set for Γ which contains eements of finite order. Show that G(Γ, S) has a Hamiton decomposition. Aspach s conjecture has aso been shown to hod when n = 1, r = 0, and the generating set S has size 2, by Bryant, Herke, Maenhaut and Webb [7]. In a paper in preparation [8], the first two authors consider the genera case when n = 1 and the underying Cayey graph is 4-reguar. Since the Cayey graph is 2-ended, it can happen for parity reasons that no Hamiton decomposition eists. However, this is the ony obstruction, and in a other cases the Cayey graphs have a Hamiton decomposition. Together with the resut of Bermond, Favaron and Maheo [6] for finite abeian groups, and the case Γ = (Z 2, +) of Theorem 1.1, this fuy characterises the 4-reguar connected Cayey graphs of finite abeian groups which have Hamiton decompositions. A natura net step woud be to consider the case of 6-reguar Cayey graphs. Probem 2. Let Γ be a finitey generated abeian group and et S be a generating set of Γ such that C(Γ, S) is 6-reguar. Characterise the pairs (Γ, S) such that G(Γ, S) has a decomposition into spanning doube-rays.

17 HAMILTON DECOMPOSITIONS OF ONE-ENDED CAYLEY GRAPHS 17 References [1] B. Aspach. Unsoved probem 4.5. Annas of Discrete Mathematics, 27:464, [2] B. Aspach. The wonderfu waecki construction. Bu. Inst. Combin. App, 52:7 20, [3] B. Aspach, J. C. Bermond, and D. Sotteau. Decomposition into cyces I: Hamiton decompositions. In Cyces and rays, pages Springer, [4] Christopher D. Aspach, B.and Godsi. Cyces in graphs, voume 27. Esevier, [5] J. C. Bermond. Hamitonian decompositions of graphs, directed graphs and hypergraphs. Annas of Discrete Mathematics, 3:21 28, [6] J. C. Bermond, O. Favaron, and M. Maheo. Hamitonian decomposition of cayey graphs of degree 4. Journa of Combinatoria Theory, Series B, 46(2): , [7] D. Bryant, S. Herke, B. Maenhaut, and B. Webb. On hamiton decompositions of infinite circuant graphs. arxiv preprint arxiv: , [8] J. Erde and F. Lehner. Hamiton decompositions of infinite 4-reguar cayey graphs. In preparation. [9] L. Fuchs. Abeian groups. Springer, [10] A. Kotzig. Every cartesian product of two circuits is decomposabe into two hamitonian circuits. Centre de Recherche Math ematiques, Montrea, [11] J. Liu. Hamitonian decompositions of cayey graphs on abeian groups. Discrete Mathematics, 131(1-3): , [12] C. Nash-Wiiams. Abeian groups, graphs and generaized knights. In Mathematica Proceedings of the Cambridge Phiosophica Society, voume 55, pages Cambridge Univ Press, [13] P. Scott and T. Wa. Topoogica methods in group theory. In Homoogica group theory (Proc. Sympos., Durham, 1977), voume 36, pages , [14] R. Stong. Hamiton decompositions of cartesian products of graphs. Discrete Mathematics, 90(2): , [15] E. Westund. Hamiton decompositions of certain 6-reguar cayey graphs on abeian groups with a cycic subgroup of inde two. Discrete Mathematics, 312(22): , [16] D. Witte and J. Gaian. A survey: Hamitonian cyces in cayey graphs. Discrete Mathematics, 51(3): , University of Hamburg, Department of Mathematics, Bundesstraße 55 (Geomatikum), Hamburg, Germany E-mai address: joshua.erde@uni-hamburg.de University of Warwick, Mathematics Institute, Zeeman Buiding, Coventry CV4 7AL, United Kingdom E-mai address: mai@forian-ehner.net University of Hamburg, Department of Mathematics, Bundesstraße 55 (Geomatikum), Hamburg, Germany E-mai address: ma.pitz@uni-hamburg.de