### Cubic graphs with large circumference deficit [article]

Edita Máčajová, Ján Mazák
<span title="2013-11-09">2013</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
The circumference c(G) 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 c(G)/|V(G)| 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. In addition, we construct snarks with large
more &raquo; ... irth and large circumference deficit, solving Problem 1 proposed in [J. Hägglund and K. Markström, On stable cycles and cycle double covers of graphs with large circumference, Disc. Math. 312 (2012), 2540--2544].
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