On Hamilton decompositions of infinite circulant graphs

Abstract

The natural infinite analog of a (finite) Hamilton cycle is a two-way-infinite Hamilton path (connected spanning 2-valent subgraph). Although it is known that every connected 2 -valent infinite circulant graph has a two-wayinfinite Hamilton path, there exist many such graphs that do not have a decomposition into edge-disjoint twoway-infinite Hamilton paths. This contrasts with the finite case where it is conjectured that every 2 -valent connected circulant graph has a decomposition into edge-disjoint Hamilton cycles. We settle the problem of decomposing 2 -valent infinite circulant graphs into edge-disjoint twoway-infinite Hamilton paths for = 2, in many cases when = 3, and in many other cases including where the connection set is ±{1, 2, … , } or ±{1, 2, … , − 1, + 1}. K E Y W O R D Scirculant graph, hamilton decomposition, infinite graph

Decompositions of complete multigraphs into cycles of varying lengths

“…The following lemma is from [8] (see Lemma 2.3 of [8] and its proof), also see [18]. Proof By Lemma 3.1 of [17], there is a decomposition D of {3, 4, .…”

Hamilton decompositions of one-ended Cayley graphs

“…Alspach's conjecture has also been shown to hold 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 general case when n = 1 and the underlying Cayley graph is 4-regular.…”

“…In particular, since each Hamiltonian double-ray must meet every edge cut an odd number of times, there can be parity reasons why no decomposition exists. One particular two-ended case, namely where Γ ∼ = Z, has been considered by Bryant, Herke, Maenhaut and Webb [7], who showed that when G(Z, S) is 4-regular, then G has a Hamilton decomposition unless there is an odd cut separating the two ends.…”

On factorisations of complete graphs into circulant graphs and the Oberwolfach problem

“…We now obtain results on 2-factorisations of Cay(Z n ; ±{1, 2, 3}), but first we need some def- For m ≥ 12 and even We can obtain an analogue of Lemma 18 for Cay(Z n ; ±{1, 3, 4}) by using using similar methods, but we will require F to have girth at least 6. The graph with vertex set {0, Lemmas 19 and 20 allow us to obtain 2-factorisations of Cay(Z n ; ±{1, 3, 4}) via the same method we used in the case of Cay(Z n ; ±{1, 2, 3}), providing we can find appropriate decomposi- 3,4,7,8,5,9,10,6) ∪ (11,12,13,17,16,19,15,18,14) For m ≥ 18 and even…”

“…• H 1 consists of the 8-cycle (0, 1,5,6,2,3,7,4) • H 2 consists of the 8-cycle (1,2,5,9,6,7,8,4) • H 2 consists of the 8-cycle (1,4,8,12,9,6,2,5) (3,4,7,8,5,9,10,6)…”