Double-critical graph conjecture for claw-free graphs

A connected graph G with chromatic number t is double-critical if G∖{x,y} is (t−2)-colorable for each edge xy∈E(G). The complete graphs are the only known examples of double-critical graphs. A long-standing conjecture of Erdős and Lovász 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≤5, but remains open for t≥6. In this paper, we first prove that if G is a non-complete, double-critical graph with chromatic number t≥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≤8 if G is claw-free.