Decomposition of $(2k+1)$-regular graphs containing special spanning $2k$-regular Cayley graphs into paths of length $2k+1$

TL;DR

This paper proves a conjecture about decomposing certain regular graphs into paths of specific lengths, focusing on graphs with particular spanning subgraphs like Cayley graphs and powers of cycles.

Contribution

It verifies the Favaron-Genest-Kouider conjecture for (2k+1)-regular graphs containing the kth power of a spanning cycle and for 5-regular graphs with special spanning Cayley graphs.

Findings

Verified the conjecture for graphs containing the kth power of a spanning cycle.

Confirmed the conjecture for 5-regular graphs with special spanning Cayley graphs.

Extended the class of graphs known to admit such path decompositions.

Abstract

A $P_\ell$-decomposition of a graph $G$ is a set of paths with $\ell$ edges in $G$ that cover the edge set of $G$. Favaron, Genest, and Kouider (2010) conjectured that every $(2k+1)$-regular graph that contains a perfect matching admits a $P_{2k+1}$-decomposition. They also verified this conjecture for $5$-regular graphs without cycles of length $4$. In 2015, Botler, Mota, and Wakabayashi verified this conjecture for $5$-regular graphs without triangles. In this paper, we verify it for $(2k+1)$-regular graphs that contain the $k$th power of a spanning cycle; and for $5$-regular graphs that contain special spanning $4$-regular Cayley graphs.