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arXiv · 1701.08506

On Hamilton Decompositions of Infinite Circulant Graphs

Darryn Bryant·Sarada Herke·Barbara Maenhaut·Bridget Webb
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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 $\pm\{1,2,\ldots,k-1,k+1\}$.

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Darryn Bryant, Sarada Herke, Barbara Maenhaut, Bridget Webb. 2017-01-30. On Hamilton Decompositions of Infinite Circulant Graphs. https://arxiv.org/abs/1701.08506

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