A note on M2-edge colorings of graphs

Abstract

An edge coloring $\varphi$ of a graph $G$ is called an $M_2$-edge coloring if $|\varphi(v)|\le2$ for every vertex $v$ of $G$, where $\varphi(v)$ is the set of colors of edges incident with $v$. Let $K_{2}(G)$ denote the maximum number of colors used in an $M_2$-edge coloring of $G$. Let $G_1$, $G_2$ and $G_3$ be graphs such that $G_1\subseteq G_2\subseteq G_3$. In this paper we deal with the following question: Assuming that $K_2(G_1)=K_2(G_3)$, does it hold $K_2(G_1)=K_2(G_2)=K_2(G_3)$?