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

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 sufficient 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 $H_{x,y}$, where $H$ is any snark and $x,y$ any pair of nonadjacent vertices of $H.$ For such superpositioned snarks, two sufficient 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 Liu et al. (2021) 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.