### Small Snarks with Large Oddness

#### 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 $5.41\, \omega(G)$ 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 $46$.