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2013 ON $r$-EQUITABLE COLORING OF COMPLETE MULTIPARTITE GRAPHS
Chih-Hung Yen
Taiwanese J. Math. 17(3): 991-998 (2013). DOI: 10.11650/tjm.17.2013.2666

Abstract

Let $r \geqslant 0$ and $k \geqslant 1$ be integers. We say that a graph $G$ has an $r$-equitable $k$-coloring if there exists a proper $k$-coloring of $G$ such that the sizes of any two color classes differ by at most $r$. The least $k$ such that a graph $G$ has an $r$-equitable $k$-coloring is denoted by $\chi_{r=}(G)$, and the least $n$ such that a graph $G$ has an $r$-equitable $k$-coloring for all $k \geqslant n$ is denoted by $\chi^*_{r=}(G)$. In this paper, we propose a necessary and sufficient condition for a complete multipartite graph $G$ to have an $r$-equitable $k$-coloring, and also give exact values of $\chi_{r=}(G)$ and $\chi^*_{r=}(G)$.

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Chih-Hung Yen. "ON $r$-EQUITABLE COLORING OF COMPLETE MULTIPARTITE GRAPHS." Taiwanese J. Math. 17 (3) 991 - 998, 2013. https://doi.org/10.11650/tjm.17.2013.2666

Information

Published: 2013
First available in Project Euclid: 10 July 2017

zbMATH: 1280.05048
MathSciNet: MR3072273
Digital Object Identifier: 10.11650/tjm.17.2013.2666

Subjects:
Primary: 05C15

Keywords: $r$-equitable chromatic number , $r$-equitable chromatic threshold , $r$-equitable coloring , complete multipartite graph , equitable coloring

Rights: Copyright © 2013 The Mathematical Society of the Republic of China

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Vol.17 • No. 3 • 2013
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