Hamiltonian cycles with all small even chords

Let G be a graph of order n ≥ 3 . An even squared Hamiltonian cycle (ESHC) of G is a Hamiltonian cycle C = v 1 v 2 … v n v 1 of G with chords v i v i + 3 for all 1 ≤ i ≤ n (where v n + j = v j for j ≥ 1 ). When n is even, an ESHC contains all bipartite 2 -regular graphs of order n . We prove that there is a positive integer N such that for every graph G of even order n ≥ N , if the minimum degree is δ ( G ) ≥ n 2 + 92 , then G contains an ESHC. We show that the condition of n being even cannot be dropped and the constant 92 cannot be replaced by 1 . Our results can be easily extended to even k th powered Hamiltonian cycles for all k ≥ 2 .