2024/11/02 by Couto, Fernanda, Ferraz, Diego Amaro, Klein, Sulamita
#Combinatorics (math.CO) #Discrete Mathematics (cs.DM) #FOS: Computer and information sciences #FOS: Mathematics
paper · doi:10.48550/arxiv.2411.01314
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 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 the edges of 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. The edge coloring problem is open for split graphs in general. In previous results, we classified split graphs with σ= 2 and in this paper we classify and provide an algorithm to color the edges of a subclass of split graphs with σ= 3.