Normal 5-edge coloring of some more snarks superpositioned by the Petersen graph

Abstract

A normal 5-edge-coloring of a cubic graph is a coloring such that for every edge the number of distinct colors incident to its end-vertices is 3 or 5 (and not 4). The well known Petersen Coloring Conjecture is equivalent to the statement that every bridgeless cubic graph has a normal 5-edge-coloring. All 3-edge-colorings of a cubic graph are obviously normal, so in order to establish the conjecture it is sufficient to consider only snarks. In our previous paper [J. Sedlar, R. \v{S}krekovski, Normal 5-edge-coloring of some snarks superpositioned by the Petersen graph, Applied Mathematics and Computation 467 (2024) 128493], we considered superpositions of any snark G along a cycle C by two simple supervertices and by the superedge obtained from the Petersen graph, but only for some of the possible ways of connecting supervertices and superedges. The present paper is a continuation of that paper, herein we consider superpositions by the Petersen graph for all the remaining connections and establish that for all of them the Petersen Coloring Conjecture holds.