Cyclic pseudo-{L}oupekine snarks

TL;DR

This paper generalizes Loupekine's method to construct cyclically symmetric snarks, introducing new infinite families with specific rotational symmetries and properties, expanding the understanding of snark structures.

Contribution

It presents a generalized construction of cyclic pseudo-Loupekine snarks using voltage graphs, creating new infinite families with specific symmetry and oddness properties.

Findings

Developed three infinite families of cyclic pseudo-Loupekine snarks with rotational symmetry.

Constructed a new infinite family of snarks with order 12m for odd m, starting from 3-edge-colorable graphs.

Showed that the oddness can increase with the parameter m in these families.

Abstract

In 1976, Loupekine introduced (via Isaacs) a very general way of constructing new snarks from old snarks by cyclically connecting multipoles constructed from smaller snarks. In this paper, we generalize Loupekine's construction to produce a variety of snarks which can be drawn with $m$-fold rotational symmetry for $m\geq 3$ (and often, $m$ odd), constructed as $\mathbb{Z}_{m}$ lifts of \emph{voltage graphs} with certain properties; we call these snarks \emph{cyclic pseudo-Loupekine snarks}. In particular, we discuss three infinite families of snarks which can be drawn with $\mathbb{Z}_{m}$ rotational symmetry whose smallest element is constructed from 3 snarks with 3-fold rotational symmetry on 28 vertices; one family has the property that the oddness of the family increases with $m$. We also develop a new infinite family of snarks, of order $12m$ for each odd $m\geq 3$, which can be drawn with $m$-fold rotational symmetry and which are constructed beginning with a 3-edge-colorable graph, instead of a snark.