On hamiltonian colorings of trees

Abstract

A hamiltonian coloring c of a graph G of order n is a mapping c: V (G) → {0, 1, 2,…} such that D(u, v) + |c(u) − c(v)| ≥ n − 1, for every two distinct vertices u and v of G, where D(u, v) denotes the detour distance between u and v which is the length of a longest u, v-path in G. The value hc(c) of a hamiltonian coloring c is the maximum color assigned to a vertex of G. The hamiltonian chromatic number, denoted by hc(G), is the min{hc(c)} taken over all hamiltonian coloring c of G. In this paper, we present a lower bound for the hamiltonian chromatic number of trees and give a sufficient condition to achieve this lower bound. Using this condition we determine the hamiltonian chromatic number of symmetric trees, firecracker trees and a special class of caterpillars.