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 $2le kle 6$ 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 $46$.
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机译：在给定的奇数和循环连通性的情况下，我们估计三次图的最小顶点数。我们证明了一个奇数为$ omega（G）$的无桥三次图$ G $，而不是Petersen图具有至少$ 5.41 ， omega（G）$的顶点，以及每个整数$ k $和$ 2 le k le 6 $我们构造了一个无限循环的立方图族，其循环连通性为$ k $，奇数比为$ | V（G）| / omega（G）$。特别是，对于$ 2 $，$ 4 $，$ 5 $和$ 6 $的周期性连接，我们将snarks的奇数比的上限从已知值$ 9提高到$ 7.5 $，$ 13 $，$ 25 $和$ 99 $。 $，$ 15 $，$ 76 $和$ 118 $。另外，我们构造了一个循环连接的$ 4 $连接的周长$ 5 $的蛇，在$ 44 $的顶点上形成了奇数$ 4 $，从而提高了以前的最佳值$ 46 $。
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