On Hamilton decompositions of infinite circulant graphs

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

On Alspach's conjecture with two even cycle lengths

For m, n even and n > m, the obvious necessary conditions for the existence of a decomposition of the complete graph K-v when v is odd (or the complete graph with a 1-factor removed K-v\F when v is even) into v in-cycles and s n-cycles are shown to be sufficient if and only if they are sufficient for v < 7n. This result is used to settle all remaining cases with m, n less than or equal to 10. (C) 2000 Elsevier Science B.V. All rights reserved