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arXiv · 2406.16618

Strictly critical snarks with girth or cyclic connectivity equal to 6

Ján Mazák·Jozef Rajník·Martin Škoviera
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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 $n\equiv 2\pmod8$.

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Ján Mazák, Jozef Rajník, Martin Škoviera. 2024-06-24. Strictly critical snarks with girth or cyclic connectivity equal to 6. https://arxiv.org/abs/2406.16618

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