A Chvátal–Erdős type condition for pancyclability

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