M₂-edge colorings of dense graphs

Abstract

An edge coloring $\varphi$ of a graph $G$ is called an $\mathrm{M}_i$-edge coloring if $|\varphi(v)|\leq i$ for every vertex $v$ of $G$, where $\varphi(v)$ is the set of colors of edges incident with $v$. Let $\mathcal{K}_i(G)$ denote the maximum number of colors used in an $\mathrm{M}_i$-edge coloring of $G$. In this paper we establish some bounds of $\mathcal{K}_2(G)$, present some graphs achieving the bounds and determine exact values of $\mathcal{K}_2(G)$ for dense graphs.