On Hamilton Decompositions of Infinite Circulant Graphs

TL;DR

This paper investigates Hamilton decompositions of infinite circulant graphs, establishing new results for specific valences and connection sets, contrasting finite and infinite cases.

Contribution

It proves the existence of Hamilton decompositions into edge-disjoint paths for certain infinite circulant graphs, especially for valence 4 and some cases for valence 6.

Findings

Infinite circulant graphs with valence 4 can be decomposed into two Hamilton paths.

Partial results for valence 6 graphs with specific connection sets.

Contrasts with finite circulant graph conjectures on Hamilton decompositions.

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 $\pm\{1,2,\ldots,k\}$ or $\pm\{1,2,\ldots,k-1,k+1\}$.