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= ^v1v2... ^{vn v1} of G with chords ^vivi+ ₃ for all 1≤i≤n (where vn+ _j= ^vj 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)<n2+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 kth powered Hamiltonian cycles for all k<2.