Strictly critical snarks with girth or cyclic connectivity equal to 6

TL;DR

This paper constructs new examples of strictly critical snarks with high cyclic connectivity and girth, filling gaps in the known existence of such graphs for even orders starting from specific minimal sizes.

Contribution

The paper introduces the first known cyclically 6-connected strictly critical snarks for all even orders ≥342 and cyclically 5-connected strictly critical snarks with girth 6 for even orders ≥66, expanding the catalog of such graphs.

Findings

Constructed cyclically 6-connected strictly critical snarks for all even n ≥ 342.

Constructed cyclically 5-connected strictly critical snarks with girth 6 for even n ≥ 66.

Filled gaps in the existence of strictly critical snarks with high cyclic connectivity.

Abstract

A snark -- connected cubic graph with chromatic index $4$ -- is critical if the graph resulting from the removal of any pair of distinct adjacent vertices is $3$-edge-colourable; it is bicritical if the same is true for any pair of distinct vertices. A snark is strictly critical if it is critical but not bicritical. Very little is known about strictly critical snarks. Computational evidence suggests that strictly critical snarks constitute a tiny minority of all critical snarks. Strictly critical snarks of order $n$ exist if and only if $n$ is even and at least 32, and for each such order there is at least one strictly critical snark with cyclic connectivity $4$. A sparse infinite family of cyclically $5$-connected strictly critical snarks is also known, but those with cyclic connectivity greater than $5$ have not been discovered so far. In this paper we fill the gap by constructing cyclically $6$-connected strictly critical snarks of each even order $n\ge 342$. In addition, we construct cyclically $5$-connected strictly critical snarks of girth 6 for every even $n\ge 66$ with $n\equiv 2\pmod8$.