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 special spanning tree in which the distance between any two adjacent vertices 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 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. When both, edges and vertices, are simultaneously colored, i.e., a total coloring of $G$, it is conjectured that any graph can be total colored with $\Delta+1$ or $\Delta+2$ colors, and thus can be classified as Type 1 or Type 2. These both variants are still open for split graphs. In this paper, using the partition of split graphs presented above, we consider the edge coloring problem and the total coloring problem for split graphs with $\sigma=2$. For this class, we characterize Class 2 and Type 2 graphs and we provide polynomial-time algorithms to color any Class 1 or Type 1 graph.

翻译：分裂图是指其顶点集可分割为一个团和一个独立集的图。若连通图$G$存在一棵特殊的生成树，使得任意两个相邻顶点之间的距离至多为$t$，则称$G$是$t$可容许的。给定图$G$，确定使$G$为$t$可容许的最小$t$值（即$G$的伸展指数，记为$\sigma(G)$）是$t$可容许性问题的目标。分裂图是$3$可容许的，并可划分为三个子类：$\sigma=1$、$2$或$3$的分裂图。本文在考虑分裂图着色问题时采用了这一划分方法。Vizing证明任何图都可以用$\Delta$或$\Delta+1$种颜色对其边进行着色，因此可分别归为第1类或第2类。当边和顶点同时被着色时（即对$G$进行全着色），猜想任何图都可以用$\Delta+1$或$\Delta+2$种颜色进行全着色，从而可归为第1型或第2型。这两类变体问题对分裂图而言仍然是开放问题。本文利用上述分裂图划分方法，研究$\sigma=2$的分裂图的边着色和全着色问题。针对此类图，我们刻画了第2类和第2型图的特征，并提出了适用于任意第1类或第1型图的多项式时间着色算法。