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Cubic graphs with large circumference deficit
[article]
<span title="2013-11-09">2013</span>
<i >
arXiv
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<span class="release-stage" >pre-print</span>
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<span title="2013-10-03">2013</span>
<span class="release-stage" >pre-print</span>
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/1310.1042v1">arXiv:1310.1042v1</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/etgtfnrj4vfhhlp3zd6sc2vtaq">fatcat:etgtfnrj4vfhhlp3zd6sc2vtaq</a> </span>
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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
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... 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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