K-ordered Hamiltonian Graphs

In this chapter, we review the following results proved in ^{1, 2, 3, 4}: (i) For k ≥ 3, every (k+1)-Hamiltonian-connected graph is k-ordered. Determine f(k, n) if n is sufficiently large in terms of k. Let g(k,n) = n/2 + k/2 -1 (ii) f(k,n) = g(k,n) if n ≥ 1 lk-3 (iii) f(k,n) ≥g(k,n) for any n ≥2k and f(k,n)> g(k,n)if 2k≤n≤3k-6 (iv) if G is a graph of order n with 3≤k≤n/2 and deg(u)+deg (v) ≥n+(3k-9)/2 for every pair u, v of non-adjacent vertices of G, then G is k-ordered Hamiltonian.