Improved bounds for the shortness coefficient of cyclically 4-edge connected cubic graphs and snarks

TL;DR

This paper constructs an infinite family of cyclically 4-edge connected cubic graphs and snarks with bounded cycle length proportional to the number of vertices, establishing new upper bounds for their shortness coefficient.

Contribution

It introduces a novel construction demonstrating an upper bound for the shortness coefficient in cyclically 4-edge connected cubic graphs and snarks, and analyzes their oddness growth.

Findings

Existence of cubic graphs with no cycles longer than (12/13) times the number of vertices.

Constructed graphs are snarks with linearly growing oddness.

Proved certain natural graph families cannot reduce the shortness coefficient to zero.

Abstract

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.