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 ) ≥ k n / ( k + 1 ) , then G contains the k th power C n k of a Hamiltonian cycle C n 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 K 1 , s -free if it does not contain an induced subgraph isomorphic to K 1 , 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 k th 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 K 1 , s -free graph of order n with δ ( G ) ≥ ( 1 / 2 + ϵ ) n contains a C n k .