Small snarks with large oddness

TL;DR

This paper investigates the minimum size of cubic graphs with specified oddness and cyclic connectivity, providing new bounds and constructions for snarks with large oddness ratios and specific connectivity properties.

Contribution

It establishes lower bounds on the number of vertices for cubic graphs with given oddness and cyclic connectivity, and constructs new examples of snarks with improved parameters.

Findings

Lower bound of 5.41 times oddness on vertices for certain cubic graphs

Improved upper bounds on oddness ratios for cyclic connectivity 2, 4, 5, 6

Constructed a cyclically 4-connected snark with girth 5 and oddness 4 on 44 vertices

Abstract

We estimate the minimum number of vertices of a cubic graph with given oddness and cyclic connectivity. We prove that a bridgeless cubic graph $G$ with oddness $\omega(G)$ other than the Petersen graph has at least $5.41\cdot\omega(G)$ vertices, and for each integer $k$ with $2\le k\le 6$ we construct an infinite family of cubic graphs with cyclic connectivity $k$ and small oddness ratio $|V(G)|/\omega(G)$. In particular, for cyclic connectivity 2, 4, 5, and 6 we improve the upper bounds on the oddness ratio of snarks to 7.5, 13, 25, and 99 from the known values 9, 15, 76, and 118, respectively. In addition, we construct a cyclically 4-connected snark of girth 5 with oddness 4 on 44 vertices, improving the best previous value of 46.