On hamiltonian colorings of block graphs

TL;DR

This paper investigates hamiltonian colorings of block graphs, establishing bounds, characterizations, and algorithms for optimal coloring, especially focusing on symmetric block graphs.

Contribution

It provides a lower bound for the hamiltonian chromatic number of block graphs, characterizes symmetric block graphs achieving this bound, and introduces algorithms for optimal coloring.

Findings

Established a lower bound for the hamiltonian chromatic number of block graphs.

Characterized symmetric block graphs that achieve the lower bound.

Developed two algorithms for optimal hamiltonian coloring of symmetric block graphs.

Abstract

A hamiltonian coloring c of a graph G of order p is an assignment of colors to the vertices of G such that $D(u,v)+|c(u)-c(v)|\geq p-1$ for every two distinct vertices u and v of G, where D(u,v) denoted the detour distance 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 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 block graphs and give a sufficient condition to achieve the lower bound. We characterize symmetric block graphs achieving this lower bound. We present two algorithms for optimal hamiltonian coloring of symmetric block graphs.