Improved bounds for the shortness coefficient of cyclically 4-edge connected cubic graphs and snarks
K Markström - arXiv preprint arXiv:1309.3870, 2013 - arxiv.org
arXiv preprint arXiv:1309.3870, 2013•arxiv.org
We present a construction which shows that there is an infinite set of cyclically 4-edge
connected cubic graphs on $ n $ vertices with no cycle longer than $ c_4 n $ for $ c_4=\frac
{12}{13} $, and at the same time prove that a certain natural family of cubic graphs cannot be
used to lower the shortness coefficient $ c_4 $ to 0. The graphs we construct are snarks so
we get the same upper bound for the shortness coefficient of snarks, and we prove that the
constructed graphs have an oddness growing linearly with the number of vertices.
connected cubic graphs on $ n $ vertices with no cycle longer than $ c_4 n $ for $ c_4=\frac
{12}{13} $, and at the same time prove that a certain natural family of cubic graphs cannot be
used to lower the shortness coefficient $ c_4 $ to 0. The graphs we construct are snarks so
we get the same upper bound for the shortness coefficient of snarks, and we prove that the
constructed graphs have an oddness growing linearly with the number of vertices.
We present a construction which shows that there is an infinite set of cyclically 4-edge connected cubic graphs on vertices with no cycle longer than for , and at the same time prove that a certain natural family of cubic graphs cannot be used to lower the shortness coefficient to 0. The graphs we construct are snarks so we get the same upper bound for the shortness coefficient of snarks, and we prove that the constructed graphs have an oddness growing linearly with the number of vertices.
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