Resistance, oddness and colouring defect of snarks

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 $\mu_3(G) \geq d$.