Is there some absolute ε > 0 such that for any claw-free graph G, the chromatic number of the square of G satisûes χ(G 2 ) ≤ (2 − ε)ω(G) 2 , where ω(G) is the clique number of G? Erdős and Nešetřil asked this question for the speciûc case where G is the line graph of a simple graph, and this was answered in the aõrmative by Molloy and Reed. We show that the answer to the more general question is also yes, and, moreover, that it essentially reduces to the original question of Erdős and Nešetřil.