Double-critical graph conjecture for claw-free graphs

TL;DR

This paper advances the understanding of the Double-Critical Graph Conjecture by proving it holds for claw-free graphs with chromatic number up to 8, under certain degree conditions.

Contribution

It proves the conjecture for claw-free graphs with chromatic number up to 8, extending previous results and introducing new degree-based structural constraints.

Findings

No degree t+1 vertex is adjacent to another degree t+1, t+2, or t+3 vertex in non-complete double-critical graphs with t ≥ 6.

The conjecture is confirmed for claw-free graphs with chromatic number up to 8.

Provides structural insights into the degree distribution of double-critical graphs.

Abstract

A connected graph $G$ with chromatic number $t$ is double-critical if $G \backslash \{x, y\}$ is $(t - 2)$-colorable for each edge $xy \in E(G)$. The complete graphs are the only known examples of double-critical graphs. A long-standing conjecture of Erd\H os and Lov\'asz from 1966, which is referred to as the Double-Critical Graph Conjecture, states that there are no other double-critical graphs. That is, if a graph $G$ with chromatic number $t$ is double-critical, then $G$ is the complete graph on $t$ vertices. This has been verified for $t \le 5$, but remains open for $t \ge 6$. In this paper, we first prove that if $G$ is a non-complete, double-critical graph with chromatic number $t \ge 6$, then no vertex of degree $t + 1$ is adjacent to a vertex of degree $t+1$, $t + 2$, or $t + 3$ in $G$. We then use this result to show that the Double-Critical Graph Conjecture is true for double-critical graphs $G$ with chromatic number $t \le 8$ if $G$ is claw-free.