2024/06/24 by Mazák, Ján, Rajník, Jozef, Škoviera, Martin
#05C15 #05C75 #Combinatorics (math.CO) #FOS: Mathematics
paper · doi:10.48550/arxiv.2406.16618
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≥ 342. In addition, we construct cyclically 5-connected strictly critical snarks of girth 6 for every even n≥ 66 with n≡ 2\pmod8.