Odd decompositions of eulerian graphs

TL;DR

This paper characterizes when Eulerian graphs can be decomposed into odd-length closed trails, introduces a conjecture for regular graphs, and verifies it for small degrees, with implications for signed graph flow numbers.

Contribution

It provides a necessary and sufficient condition for decomposing Eulerian graphs into odd closed trails and proposes a new conjecture for regular graphs, verified for degrees up to three.

Findings

Decomposition criterion for Eulerian graphs into odd closed trails

Verification of the conjecture for 2d-regular graphs with d ≤ 3

Connection to flow numbers of signed Eulerian graphs

Abstract

We prove that an eulerian graph $G$ admits a decomposition into $k$ closed trails of odd length if and only if and it contains at least $k$ pairwise edge-disjoint odd circuits and $k\equiv |E(G)|\pmod{2}$. We conjecture that a connected $2d$-regular graph of odd order with $d\ge 1$ admits a decomposition into $d$ odd closed trails sharing a common vertex and verify the conjecture for $d\le 3$. The case $d=3$ is crucial for determining the flow number of a signed eulerian graph which is treated in a separate paper (arXiv:1408.1703v2). The proof of our conjecture for $d=3$ is surprisingly difficult and calls for the use of signed graphs as a convenient technical tool.