Cyclic pseudo-{L}oupekine snarks
Leah Wrenn Berman, D\'eborah Oliveros, Gordon I. Williams

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 -fold rotational symmetry for (and often, odd), constructed as 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 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 . We also develop a new infinite family of snarks, of order for each odd , which can be…
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Taxonomy
Topicsgraph theory and CDMA systems · Advanced Graph Theory Research · Finite Group Theory Research
