On hamiltonian colorings of trees

TL;DR

This paper investigates hamiltonian colorings of trees, establishing a lower bound for their hamiltonian chromatic number and identifying conditions under which this bound is tight, with exact values computed for specific tree classes.

Contribution

It introduces a lower bound for the hamiltonian chromatic number of trees and provides a sufficient condition for achieving this bound, along with exact calculations for certain tree types.

Findings

Lower bound for hamiltonian chromatic number of trees

Sufficient condition for optimal hamiltonian coloring

Exact hamiltonian chromatic numbers for symmetric trees, firecracker trees, and certain caterpillars

Abstract

A hamiltonian coloring $c$ of a graph $G$ of order $n$ is a mapping $c$ : $V(G) \rightarrow \{0,1,2,...\}$ such that $D(u, v)$ + $|c(u) - c(v)|$ $\geq$ $n-1$, 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.