Resistance, oddness and colouring defect of snarks

TL;DR

This paper constructs non-trivial snarks with prescribed resistance, oddness, and defect parameters, demonstrating the unbounded variability of these measures of uncolourability in cubic graphs.

Contribution

It proves the existence of non-trivial snarks with any given resistance, oddness, and defect within specified bounds, advancing understanding of their structural properties.

Findings

Existence of snarks with arbitrary resistance and oddness ratios.

Construction of snarks with prescribed resistance, oddness, and defect.

Demonstration that defect can be arbitrarily large relative to oddness.

Abstract

Let $G$ be a bridgeless cubic graph. The \textit{resistance} of $G$, denoted $r(G)$, is the minimum number of edges which can be removed from $G$ in order to render 3-edge-colourability. The \textit{oddness} of $G$, denoted $\omega(G)$, is the minimum number of odd components in a 2-factor of $G$. The \textit{colouring defect} of $G$ (or simply, the \textit{defect} of $G$), denoted $\mu_3(G)$, is the minimum number of edges not contained in any set of three perfect matchings of $G$. These three parameters are regarded as measurements of uncolourability of snarks, partly because any one of these parameters equal zero if and only if $G$ is 3-edge-colourable. It is also known that $r(G) \geq \omega(G)$ and that $\mu_3(G) \geq \frac{3}{2}\omega(G)$ \cite{fiol,jinsteffen}. We have shown that the ratio of oddness to resistance can be arbitrarily large for non-trivial snarks \cite{allie1}. It has also been shown that the ratio of the defect to oddness can be arbitrarily large for non-trivial snarks, although this result was only shown for graphs with oddness equal to 2 \cite{karabasetal}. In the same paper, the question was posed whether there exists non-trivial snarks for given resistance $r$ or given oddness $\omega$, and arbitrarily large defect. In this paper, we prove a stronger result: For any positive integers $r \geq 2$, even $\omega \geq r$, and $d \geq \frac{3}{2}\omega$, there exists a non-trivial snark $G$ with $r(G)=r$, $\omega(G)=\omega$ and $\mu_3(G) \geq d$.