A note on powers of Hamilton cycles in generalized claw-free graphs

Abstract

Seymour conjectured for a fixed integer k≥2 that if G is a graph of order n with δ(G)≥kn/(k+1), then G contains the kth power Cnk of a Hamiltonian cycle Cn of G, and this minimum degree condition is sharp. Earlier the k=2 case was conjectured by Pósa. This was verified by Komlós et al. [4]. For s≥3, a graph is K1,s-free if it does not contain an induced subgraph isomorphic to K1,s. Such graphs will be referred to as generalized claw-free graphs. Minimum degree conditions that imply that a generalized claw-free graph G of sufficiently large order n contains a kth power of a Hamiltonian cycle will be proved. More specifically, it will be shown that for any ϵ>0 and for n sufficiently large, any K1,s-free graph of order n with δ(G)≥(1/2+ϵ)n contains a Cnk.