Weak oddness as an approximation of oddness and resistance in cubic graphs

TL;DR

The paper introduces weak oddness, a new measure for cubic graphs' uncolourability, which approximates oddness and resistance, and explores their relationships and differences through constructions.

Contribution

It defines weak oddness as an approximation of oddness and resistance, and analyzes their relationships, including cases with large differences and effects of vertex replacement.

Findings

Weak oddness bounds resistance and oddness in cubic graphs.

Existence of graphs where resistance, weak oddness, and oddness differ significantly.

Vertex replacement with a triangle can arbitrarily reduce oddness.

Abstract

We introduce weak oddness $\omega_{\textrm w}$, a new measure of uncolourability of cubic graphs, defined as the least number of odd components in an even factor. For every bridgeless cubic graph $G$, $\rho(G)\le\omega_{\textrm w}(G)\le\omega(G)$, where $\rho(G)$ denotes the resistance of $G$ and $\omega(G)$ denotes the oddness of $G$, so this new measure is an approximation of both oddness and resistance. We demonstrate that there are graphs $G$ satisfying $\rho(G) < \omega_{\textrm w}(G) < \omega(G)$, and that the difference between any two of those three measures can be arbitrarily large. The construction implies that if we replace a vertex of a cubic graph with a triangle, then its oddness can decrease by an arbitrarily large amount.