New Results on Edge-coloring and Total-coloring of Split Graphs

TL;DR

This paper investigates edge-coloring and total-coloring for $σ=2$-split graphs, a natural subclass of split graphs arising in the $t$-admissibility framework. It provides a complete characterization of Class 2 and Type 2 graphs via the neighborhood-overfull concept and delivers polynomial-time algorithms to obtain a $Δ$-edge-coloring for Class 1 and a $(Δ+1)$-total-coloring for Type 1 graphs by reducing coloring to the subgraph $H=G[V\setminus P]$ and extending to pendant vertices/edges. Key theoretical contributions include a pendant-vertex lemma, a reduction to $H$, and implications for Hilton's Conjecture and overfull-subgraph conjectures within split graphs. The work advances toward a full classification of split graphs with respect to edge and total coloring and outlines future directions to address the σ=3 case for a complete taxonomy.

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 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 $σ(G)$, is the goal of the $t$-admissibility problem. Split graphs are $3$-admissible and can be partitioned into three subclasses: split graphs with $σ=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 $Δ$ or $Δ+1$ 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 $Δ+1$ or $Δ+2$ 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 $σ=2$. 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.

Figures (5)

• Figure 1: Intersection between split graphs and $3$-admissible graphs. Note that any $t~$- admissible graph is a $k$-admissible graph for $k>t$. But the converse is not true.
• Figure 2: (a) and (b) Minimal forbidden induced subgraphs which are common to both split-interval and split-comparability graphs. (c) Minimal forbidden induced subgraph for split-interval graphs, exclusively. (d) Minimal forbidden induced subgraph for split-comparability graphs, exclusively.
• Figure 3: State of the art of the study of the classification problem in split graphs.
• Figure 4: (a) Split Graph $G$ such that $\sigma(G)=2$. (b) Subgraph $H=G[V\setminus P]$. (c) Edge Coloring of $H$. (d) Pendant edges missing coloring. (e) Edge Coloring of $G$.
• Figure 5: (a) Split graph $G$ such that $\sigma(G)=2$. (b) Subgraph $H=G[V \setminus P]$. (c) Total-coloring of $H$. (d) Pendant edges and pendant vertices missing coloring. (e) Edge-coloring of pendant edges. (f) Total-coloring of $G$.

Theorems & Definitions (18)

• Theorem 1.1
• Theorem 2.1
• Theorem 2.2
• Theorem 2.3
• Theorem 2.4
• Lemma 2.1
• Theorem 2.5
• Theorem 2.6
• Theorem 2.7
• Conjecture 2.1