Normal 5-edge-coloring of some snarks superpositioned by Flower snarks

TL;DR

This paper investigates conditions under which certain superpositioned snarks, especially those involving Flower snarks, admit a normal 5-edge-coloring, contributing to the broader context of the Petersen Coloring Conjecture.

Contribution

It provides new sufficient conditions for normal 5-edge-colorings in superpositioned snarks, particularly those involving Flower snarks, advancing understanding of the Petersen Coloring Conjecture.

Findings

Normal 5-edge-colorings exist for superpositions with hypohamiltonian snarks.

All superpositions by Flower snarks admit a normal 5-edge-coloring under certain conditions.

The class studied has a Petersen coloring, linking to previous conjectures.

Abstract

An edge e is normal in a proper edge-coloring of a cubic graph G if the number of distinct colors on four edges incident to e is 2 or 4: A normal edge-coloring of G is a proper edge-coloring in which every edge of G is normal. The Petersen Coloring Conjecture is equivalent to stating that every bridgeless cubic graph has a normal 5-edge-coloring. Since every 3-edge-coloring of a cubic graph is trivially normal, it is suficient to consider only snarks to establish the conjecture. In this paper, we consider a class of superpositioned snarks obtained by choosing a cycle C in a snark G and superpositioning vertices of C by one of two simple supervertices and edges of C by superedges Hx;y, where H is any snark and x; y any pair of nonadjacent vertices of H: For such superpositioned snarks, two suficient conditions are given for the existence of a normal 5-edge-coloring. The first condition yields a normal 5-edge-coloring for all hypohamiltonian snarks used as superedges, but only for some of the possible ways of connecting them. In particular, since the Flower snarks are hypohamiltonian, this consequently yields a normal 5-edge-coloring for many snarks superpositioned by the Flower snarks. The second sufficient condition is more demanding, but its application yields a normal 5-edge-colorings for all superpositions by the Flower snarks. The same class of snarks is considered in [S. Liu, R.-X. Hao, C.-Q. Zhang, Berge{Fulkerson coloring for some families of superposition snarks, Eur. J. Comb. 96 (2021) 103344] for the Berge-Fulkerson conjecture. Since we established that this class has a Petersen coloring, this immediately yields the result of the above mentioned paper.