Normal 5-edge-colorings of a family of Loupekhine snarks

• Published in: AKCE international journal of graphs and combinatorics Vol. ahead-of-print; no. ahead-of-print; pp. 1 - 5
• Main Authors: Ferrarini, Luca, Mazzuoccolo, Giuseppe, Mkrtchyan, Vahan
• Format: Journal Article
• Language: English
• Published: Taylor & Francis 01.09.2020; Taylor & Francis Group
• Subjects: class of snarks; Cubic graph; normal edge-coloring; Petersen coloring conjecture
• ISSN: 0972-8600; 2543-3474
• DOI: 10.1016/j.akcej.2019.12.014

In a proper edge-coloring of a cubic graph an edge uv is called poor or rich, if the set of colors of the edges incident to u and v contains exactly three or five colors, respectively. An edge-coloring of a graph is normal, if any edge of the graph is either poor or rich. In this note, we show that some snarks constructed by using a method introduced by Loupekhine admit a normal edge-coloring with five colors. The existence of a Berge-Fulkerson Covering for a part of the snarks considered in this paper was recently proved by Manuel and Shanthi (2015). Since the existence of a normal edge-coloring with five colors implies the existence of a Berge-Fulkerson Covering, our main theorem can be viewed as a generalization of their result.