On pancyclic claw-free graphs (Q2713659)

Let \(G=(V,E)\) be a graph. A vertex \(x\in V(G)\) is said to be \(N_2\)-locally connected if the set of all edges \(uv\) with \(\min \{\)dist\((u,x), \)dist\((v,x)\}=1\) induces a connected graph. The main result shows that if \(G\) is a claw-free graph with minimum degree at least three which does not contain certain simple forbidden substructures and in which every cutset contains an \(N_2\)-locally connected vertex, then every \(N_2\)-locally connected vertex of \(G\) is contained in cycles of all possible lengths.