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

## TL;DR

This paper demonstrates that certain Loupekhine snarks, which are cubic graphs, admit a normal 5-edge-coloring, extending previous results related to Berge-Fulkerson Coverings.

## Contribution

It shows that some Loupekhine-constructed snarks have a normal 5-edge-coloring, generalizing prior findings on Berge-Fulkerson Coverings.

## Key findings

- Existence of normal 5-edge-colorings for specific Loupekhine snarks
- Connection between normal edge-colorings and Berge-Fulkerson Coverings
- Extension of previous results to a broader class of snarks

## Abstract

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.