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 $\sigma = 3$.

翻译：分裂图是指其顶点集可划分为一个团和一个独立集的图。连通图$G$若存在生成树使得$G$中任意相邻顶点在该树中的距离不超过$t$，则称该图是$t$-可容许的。对于给定图$G$，确定使其成为$t$-可容许图的最小$t$值（记为$\sigma(G)$，称为$G$的伸缩指数）是$t$-可容许性问题的核心目标。分裂图具有$3$-可容许性，并可划分为三个子类：$\sigma = 1$、$2$或$3$的分裂图。本研究基于该划分探讨分裂图的边着色问题。Vizing证明任意图可用$\Delta$或$\Delta+1$种颜色对其边进行着色，相应地图可归类为Class $1$或Class $2$。分裂图的边着色问题在一般情况下尚未完全解决。前期研究中我们已对$\sigma = 2$的分裂图完成分类，本文将进一步对$\sigma = 3$的分裂图子类进行分类，并提供相应的边着色算法。