Hamiltonian chromatic number of block graphs

Let G be a simple connected graph of order n. A hamiltonian coloring c of a graph G is an assignment of colors (non-negative 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, where D(u, v) denotes the detour distance between u and v in G which is the length of the longest path between u and v. 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 min{hc(c)} taken over all hamiltonian coloring c of G. In this paper, we give a necessary and sufficient condition to achieve a lower bound for the hamiltonian chromatic number of block graphs given in [1, Theorem 1]. We present an algorithm for optimal hamiltonian coloring of a special class of block graphs, namely SDB(p/2) block graphs. We characterize level-wise regular block graphs and extended star of blocks achieving this lower bound. Submitted: May 2016 Reviewed: November 2016 Revised: December 2016 Reviewed: December 2016 Revised: December 2016 Accepted: December 2016 Final: January 2017 Published: February 2017 Article type: Regular paper Communicated by: M. Kaykobad and R. Petreschi E-mail address: [email protected] (Devsi Bantva) 354 Devsi Bantva Hamiltonian chromatic number of block graphs