Taiwanese Journal of Mathematics


Chih-Hung Yen

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

First available in Project Euclid: 10 July 2017

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Zentralblatt MATH identifier

Primary: 05C15: Coloring of graphs and hypergraphs

equitable coloring $r$-equitable coloring complete multipartite graph $r$-equitable chromatic number $r$-equitable chromatic threshold


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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