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

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.