Harmonious Coloring of Trees with Large Maximum Degree

Details Harmonious Coloring of Trees with Large Maximum Degree Harmonious Coloring of Trees with Large Maximum Degree

2012-02-06

arxiv.org/abs/1202.1046v1

Mathematics

Combinatorics

8 pages, 1 figure

Scientific paper

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.

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