On hamiltonian colorings for some graphs

Abstract

For a connected graph G and any two vertices u and v in G, let D(u,v) denote the length of a longest u–v path in G. A hamiltonian coloring of a connected graph G of order n is an assignment c of colors (positive integers) to the vertices of G such that |c(u)−c(v)|+D(u,v)≥n−1 for every two distinct vertices u and v in G. The valuehc(c) of a hamiltonian coloring c is the maximum color assigned to a vertex of G. The hamiltonian chromatic numberhc(G) of G is min{hc(c)} taken over all hamiltonian colorings c of G. In this paper we discuss the hamiltonian chromatic number of graphs G with max{D(u,v)|u,v∈V(G),u≠v}≤n2. As examples, we determine the hamiltonian chromatic number for a class of caterpillars, and double stars.