On proper Hamiltonian-connection number of graphs

Abstract

A graph G is Hamiltonian-connected if every two vertices of G are connected by a Hamilton path. A bipartite graph H is Hamiltonian-laceable if any two vertices from different partite sets of H are connected by a Hamilton path. An edge-coloring (adjacent edges may receive the same color) of a Hamiltonian-connected (respectively, Hamiltonian-laceable) graph G (resp. H) is a proper Hamilton path coloring if every two vertices u and v of G (resp. H) are connected by a Hamilton path Puv such that no two adjacent edges of Pᵤᵥ are colored the same. The minimum number of colors in a proper Hamilton path coloring of G (resp. H) is the proper Hamiltonian-connection number of G (resp. H). In this paper, proper Hamiltonian-connection numbers are determined for some classes of Hamiltonian-connected graphs and that of Hamiltonian-laceable graphs.