An infinite family of normal 5-edge colorable superpositioned snarks

Abstract

Let $G$ be a cubic graph. For a proper edge coloring $\pi$ of $G$, an edge $e\in E\left(G\right)$ is normal if the number of colors of all edges incident to endvertices of $e$ is 3 or 5. A normal $k$-edge coloring of $G$ is a proper edge coloring with $k$ colors such that each edge of $G$ is normal. The normal chromatic index of $G$, denoted by ${\chi }_N^{\prime }\left(G\right)$, is the smallest integer $k$ such that $G$ admits a normal $k$-edge coloring. Jaeger conjectured that ${\chi }_N^{\prime }\left(G\right)\le 5$ for every bridgeless cubic graph $G$, and he also proved that this conjecture is equivalent to the Petersen coloring conjecture. The normal 5-edge coloring conjecture has been verified for some families of snarks, such as, generalized Blanuša snarks, Goldberg snarks, flower snarks and Loupekhine snarks. Sedlar et al. proved that this conjecture holds for two families of superpositioned snarks. In this paper, we present a construction of superpositioned snarks whose normal chromatic indices equal to 5, which implies the normal 5-edge coloring conjecture for large families of snarks.