## Taiwanese Journal of Mathematics

### THE EQUITABLE COLORINGS OF KNESER GRAPHS

#### Abstract

An $m$-coloring of a graph $G$ is a mapping $f:V(G)\rightarrow \{1,2,\ldots,m\}$ such that $f(x)\neq f(y)$ for any two adjacent vertices $x$ and $y$ in $G$. The chromatic number $\chi(G)$ of $G$ is the minimum number $m$ such that $G$ is $m$-colorable. An equitable $m$-coloring of a graph $G$ is an $m$-coloring $f$ such that any two color classes differ in size by at most one. The equitable chromatic number $\chi_{_=}(G)$ of $G$ is the minimum number $m$ such that $G$ is equitably $m$-colorable. The equitable chromatic threshold $\chi_{_=}^*(G)$ of $G$ is the minimum number $m$ such that $G$ is equitably $r$-colorable for all $r\geq m$. It is clear that $\chi(G)\leq \chi_{_=}(G) \leq \chi_{_=}^*(G)$. For $n\geq 2k+1$, the Kneser graph $\sf KG(n,k)$ has the vertex set consisting of all $k$-subsets of an $n$-set. Two distinct vertices are adjacent in $\sf KG(n,k)$ if they have empty intersection as subsets. The Kneser graph $\sf KG(2k+1,k)$ is called the Odd graph, denoted by $O_k$. In this paper, we study the equitable colorings of Kneser graphs $\sf KG(n,k)$. Mainly, we obtain that $\chi_{_=}(\sf KG(n,k))\leq \chi_{_=}^*(\sf KG(n,k))\leq n-k+1$ and $\chi(O_k)=\chi_{_=}(O_k)=\chi_{_=}^*(O_k)=3$. We also show that $\chi_{_=}(\sf KG(n,k))=\chi_{_=}^*(\sf KG(n,k))$ for $k=2$ or 3 and obtain their exact values.

#### Article information

Source
Taiwanese J. Math., Volume 12, Number 4 (2008), 887-900.

Dates
First available in Project Euclid: 18 July 2017

https://projecteuclid.org/euclid.twjm/1500404984

Digital Object Identifier
doi:10.11650/twjm/1500404984

Mathematical Reviews number (MathSciNet)
MR2426534

Zentralblatt MATH identifier
1168.05020

Subjects
Primary: 05C15: Coloring of graphs and hypergraphs

#### Citation

Chen, Bor-Liang; Huang, Kuo-Ching. THE EQUITABLE COLORINGS OF KNESER GRAPHS. Taiwanese J. Math. 12 (2008), no. 4, 887--900. doi:10.11650/twjm/1500404984. https://projecteuclid.org/euclid.twjm/1500404984

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