Even-cycle decompositions of graphs with no odd-$K_4$-minor

TL;DR

This paper establishes that 2-connected loopless Eulerian graphs without odd-$K_4$-minors and with an even number of edges can be decomposed into even cycles, extending previous results to a broader class of graphs.

Contribution

It proves a new sufficient condition for even-cycle decompositions in graphs excluding odd-$K_4$-minors, generalizing earlier theorems for planar and $K_5$-minor-free graphs.

Findings

Graphs without odd-$K_4$-minors have even-cycle decompositions under specified conditions.

The result is optimal; replacing 'odd-$K_4$-minor-free' with 'odd-$K_5$-minor-free' does not hold.

Structural characterization of odd-$K_4$-minor-free graphs is key to the proof.

Abstract

An even-cycle decomposition of a graph G is a partition of E(G) into cycles of even length. Evidently, every Eulerian bipartite graph has an even-cycle decomposition. Seymour (1981) proved that every 2-connected loopless Eulerian planar graph with an even number of edges also admits an even-cycle decomposition. Later, Zhang (1994) generalized this to graphs with no $K_5$-minor. Our main theorem gives sufficient conditions for the existence of even-cycle decompositions of graphs in the absence of odd minors. Namely, we prove that every 2-connected loopless Eulerian odd-$K_4$-minor-free graph with an even number of edges has an even-cycle decomposition. This is best possible in the sense that `odd-$K_4$-minor-free' cannot be replaced with `odd-$K_5$-minor-free.' The main technical ingredient is a structural characterization of the class of odd-$K_4$-minor-free graphs, which is due to Lov\'asz, Seymour, Schrijver, and Truemper.