A Chvátal–Erdős type condition for pancyclability

AbstractLet G be a graph and S a subset of V(G). Let α(S) denote the maximum number of pairwise nonadjacent vertices in the subgraph G〈S〉 of G induced by S. If G〈S〉 is not complete, let κ(S) denote the smallest number of vertices separating two vertices of S and κ(S)=|S|-1 otherwise. We prove that if α(S)⩽κ(S) and |S| is large enough (depending on α(S)), then G is S-pancyclable, that is contains cycles with exactly p vertices of S for every p, 3⩽p⩽|S|