Hamiltonian chromatic number of trees

TL;DR

This paper investigates the hamiltonian chromatic number of trees, providing an improved lower bound, necessary and sufficient conditions for optimality, and exact values for specific tree families.

Contribution

It introduces an improved lower bound for the hamiltonian chromatic number of trees and characterizes when this bound is achieved, along with exact calculations for certain tree classes.

Findings

Established a new lower bound for the hamiltonian chromatic number of trees.

Provided necessary and sufficient conditions for trees to attain this bound.

Determined the hamiltonian chromatic number for two specific families of trees.

Abstract

Let $G$ be a simple finite connected graph of order $n$. The detour distance between two distinct vertices $u$ and $v$ denoted by $D(u,v)$ is the length of a longest $uv$-path in $G$. A hamiltonian coloring $h$ of a graph $G$ of order $n$ is a mapping $h : V(G) \rightarrow \{0,1,2,...\}$ such that $D(u,v) + |h(u)-h(v)| \geq n-1$, for every two distinct vertices $u$ and $v$ of $G$. The span of $h$, denoted by $span(h)$, is $\max\{|h(u)-h(v)| : u, v \in V(G)\}$. The hamiltonian chromatic number of $G$ is defined as $hc(G) := \min\{span(h)\}$ with minimum taken over all hamiltonian coloring $h$ of $G$. In this paper, we give an improved lower bound for the hamiltonian chromatic number of trees and give a necessary and sufficient condition to achieve the improved lower bound. Using this result, we determine the hamiltonian chromatic number of two families of trees.