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

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 Ck n 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 clawfree 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 nwith δ(G) ≥ (1/2 + ε)n contains a Ck n . © 2012 Elsevier B.V. All rights reserved.