Harmonious Coloring of Trees with Large Maximum Degree

TL;DR

This paper determines the exact harmonious chromatic number for certain forests with large maximum degree and provides a polynomial-time algorithm for optimal harmonious coloring in these cases.

Contribution

It establishes the precise harmonious chromatic number for forests with large maximum degree and introduces an efficient algorithm for optimal coloring.

Findings

Harmonious chromatic number equals Δ(T)+2 if non-adjacent vertices of degree Δ(T) exist.

Harmonious chromatic number equals Δ(T)+1 otherwise.

Provides a polynomial-time algorithm for optimal harmonious coloring.

Abstract

A harmonious coloring of $G$ is a proper vertex coloring of $G$ such that every pair of colors appears on at most one pair of adjacent vertices. The harmonious chromatic number of $G$, $h(G)$, is the minimum number of colors needed for a harmonious coloring of $G$. We show that if $T$ is a forest of order $n$ with maximum degree $\Delta(T)\geq \frac{n+2}{3}$, then $$h(T)= \Delta(T)+2, & if $T$ has non-adjacent vertices of degree $\Delta(T)$; \Delta(T)+1, & otherwise. $$ Moreover, the proof yields a polynomial-time algorithm for an optimal harmonious coloring of such a forest.