A Note on Harmonious Coloring of Caterpillars

Asahi TAKAOKA  Shingo OKUMA  Satoshi TAYU  Shuichi UENO  

IEICE TRANSACTIONS on Information and Systems   Vol.E98-D   No.12   pp.2199-2206
Publication Date: 2015/12/01
Publicized: 2015/08/28
Online ISSN: 1745-1361
DOI: 10.1587/transinf.2015EDP7113
Type of Manuscript: PAPER
Category: Fundamentals of Information Systems
caterpillars,  Eulerian trail,  harmonious coloring,  harmonious chromatic number,  pathwidth,  

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The harmonious coloring of an undirected simple graph is a vertex coloring such that adjacent vertices are assigned different colors and each pair of colors appears together on at most one edge. The harmonious chromatic number of a graph is the least number of colors used in such a coloring. The harmonious chromatic number of a path is known, whereas the problem to find the harmonious chromatic number is NP-hard even for trees with pathwidth at most 2. Hence, we consider the harmonious coloring of trees with pathwidth 1, which are also known as caterpillars. This paper shows the harmonious chromatic number of a caterpillar with at most one vertex of degree more than 2. We also show the upper bound of the harmonious chromatic number of a 3-regular caterpillar.