Strictly critical snarks with girth or cyclic connectivity equal to 6

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 $n\equiv 2\left(\mathrm{mod}8\right)$.