On Hamilton decompositions of infinite circulant graphs

Bryant, Darryn; Herke, Sarada; Maenhaut, Barbara and Webb, Bridget S. (2017). On Hamilton decompositions of infinite circulant graphs. Journal of Graph Theory, 88(3) pp. 434–448.

DOI: https://doi.org/10.1002/jgt.22223

Abstract

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}.

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