Taiwanese Journal of Mathematics

ON $r$-EQUITABLE COLORING OF COMPLETE MULTIPARTITE GRAPHS

Chih-Hung Yen

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 $\chi^*_{r=}(G)$.

Article information

Source
Taiwanese J. Math., Volume 17, Number 3 (2013), 991-998.

Dates
First available in Project Euclid: 10 July 2017

https://projecteuclid.org/euclid.twjm/1499705995

Digital Object Identifier
doi:10.11650/tjm.17.2013.2666

Mathematical Reviews number (MathSciNet)
MR3072273

Zentralblatt MATH identifier
1280.05048

Subjects
Primary: 05C15: Coloring of graphs and hypergraphs

Citation

Yen, Chih-Hung. ON $r$-EQUITABLE COLORING OF COMPLETE MULTIPARTITE GRAPHS. Taiwanese J. Math. 17 (2013), no. 3, 991--998. doi:10.11650/tjm.17.2013.2666. https://projecteuclid.org/euclid.twjm/1499705995

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