On the hamiltonian path graph of a graph

The hamiltonian path graph H(F) of a graph F is that graph having the same vertex set as F and in which two vertices u and v are adjacent if and only if F contains a hamiltonian u − v path. First, in response to a conjecture of Chartrand, Kapoor and Nordhaus, a characterization of nonhamiltonian graphs isomorphic to their hamiltonian path graphs is presented. Next, the maximum size of a hamiltonian graph F of given order such that K̄ d ⊆ H(F) is determined. Finally, it is shown that if the degree sum of the endvertices of a hamiltonian path in a graph F with at least five vertices is at least | V(F) | + t(t ⩾ 0), then H(F) contains a complete subgraph of order t + 4.