Filling some gaps on the edge coloring problem of split graphs

Abstract

A split graph is a graph whose vertex set can be partitioned into a clique and an independent set. A connected graph $G$ is said to be $t$-admissible if admits a spanning tree in which the distance between any two adjacent vertices of $G$ is at most $t$. Given a graph $G$, determining the smallest $t$ for which $G$ is $t$-admissible, i.e., the stretch index of $G$ denoted by $\sigma(G)$, is the goal of the $t$-admissibility problem. Split graphs are $3$-admissible and can be partitioned into three subclasses: split graphs with $\sigma = 1$, $2$ or $3$. In this work we consider such a partition while dealing with the problem of coloring the edges of a split graph. Vizing proved that any graph can have its edges colored with $\Delta$ or $\Delta+1$ colors, and thus can be classified as Class $1$ or Class $2$, respectively. The edge coloring problem is open for split graphs in general. In previous results, we classified split graphs with $\sigma = 2$ and in this paper we classify and provide an algorithm to color the edges of a subclass of split graphs with $\sigma = 3$.