Abstract

The circumference cG of a graph G is the length of a longest cycle. By exploiting our recent results on resistance of snarks, we construct infinite classes of cyclically 4-, 5-, and 6-edge-connected cubic graphs with circumference ratio cG/|VG| bounded from above by 0.876, 0.960, and 0.990, respectively. In contrast, the dominating cycle conjecture implies that the circumference ratio of a cyclically 4-edge-connected cubic graph is at least 0.75. Up to our knowledge, no upper bounds on this ratio have been known before for cubic graphs with cyclic edge-connectivity above 3. In addition, we construct snarks with large girth and large circumference deficit, solving Problem 1 proposed in [J. Hagglund and K. Markstrom, On stable cycles and cycle double covers of graphs with large circumference, Disc Math 312 2012, 2540-2544].

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