Hamiltonian colorings of graphs

Abstract

For vertices u and v in a connected graph G of order n, the length of a longest u – v path in G is denoted by D ( u , v ) . A hamiltonian coloring c of G is an assignment c of colors (positive integers) to the vertices of G such that D ( u , v ) + | c ( u ) - c ( v ) | ⩾ n - 1 for every two distinct vertices u and v of G. The value hc ( c ) of a hamiltonian coloring c of G is the maximum color assigned to a vertex of G. The hamiltonian chromatic number hc ( G ) of G is min { hc ( c ) } over all hamiltonian colorings c of G. Hamiltonian chromatic numbers of some special classes of graphs are determined. It is shown that for every two integers k and n with k ⩾ 1 and n ⩾ 3 , there exists a hamiltonian graph of order n with hamiltonian chromatic number k if and only if 1 ⩽ k ⩽ n - 2 . Also, a sharp upper bound for the hamiltonian chromatic number of a connected graph in terms of its order is established.