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, i.e., if a graph G with chromatic number t is double-critical, then G = Kt. 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.