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 $σ(G)$, 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 $σ= 3$.

Figures (7)

• Figure 1: Relation between split graphs and $3$-admissible graphs.
• Figure 2: Some subclasses of split graphs and their relation with the $t$-admissibility problem.
• Figure 3: On the left there are two conflicting edges: $wx$, $wy$ colored with color $1$. On the right, $wx$ remains colored with color $1$ and $wy$ and $ya$ swapped colors.
• Figure 4: (a) A Color Trail that shows two color conflicts on vertex $w$. (b) First swap of colors between edges $wx_2$ and $x_2x_1'$.
• Figure 5: The next conflict treated by the algorithm is the one involving the edges $wx_2$ and $wx_3$. Note that, for this second conflict, two swaps are needed.

Theorems & Definitions (31)

• Theorem 1.1: Couto
• Theorem 2.1: Behzad
• Theorem 2.2: Behzad
• Theorem 2.3: Plantholt's Result Plantholt
• Theorem 2.4: Chen
• Theorem 2.5: Sheila
• Lemma 2.1
• proof
• Remark 2.1
• Remark 3.1