Cubic graphs with large circumference deficit
Discrete Mathematics
2013-11-12 v2 Combinatorics
Abstract
The circumference of a graph is the length of a longest cycle. By exploiting our recent results on resistance of snarks, we construct infinite classes of cyclically -, - and -edge-connected cubic graphs with circumference ratio bounded from above by , and , respectively. In contrast, the dominating cycle conjecture implies that the circumference ratio of a cyclically -edge-connected cubic graph is at least . In addition, we construct snarks with large girth and large circumference deficit, solving Problem 1 proposed in [J. H\"agglund and K. Markstr\"om, On stable cycles and cycle double covers of graphs with large circumference, Disc. Math. 312 (2012), 2540--2544].
Keywords
Cite
@article{arxiv.1310.1042,
title = {Cubic graphs with large circumference deficit},
author = {Edita Máčajová and Ján Mazák},
journal= {arXiv preprint arXiv:1310.1042},
year = {2013}
}