Sparse claw-free graph enumeration

Determine the asymptotics of the logarithm of the number of claw-free graphs on n vertices with cn edges, for each fixed constant c>0, in order to characterize the sparse component appearing in typical subcritical claw-free graphs.

Background

The paper determines the typical structure of dense claw-free graphs with fixed edge density. In the subcritical regime, almost every such graph is the vertex-disjoint union of a large co-bipartite graph and a sparse claw-free graph G_2. The proof only uses cubic claw-free graphs to lower-bound the number of possibilities for G_2, so the authors identify the enumeration of sparse claw-free graphs as a natural unresolved problem.

Solving this problem would provide a more precise description of the sparse component and could improve the asymptotic understanding of subcritical claw-free graphs.