Coloring Reduced Kneser Graph

• Main Author: 李渭天
• Other Authors: 李國偉
• Format: Others
• Language: en_US
• Published: 2003
• Online Access: http://ndltd.ncl.edu.tw/handle/71215543441167987572

碩士 === 國立臺灣大學 === 數學研究所 === 91 === The vertex set of a Kneser graph ${\sf KG}(m,n)$ consists of all $n$-subsets of the set $[m]=\{0, 1, \ldots, m-1\}$. Two vertices are defined to be adjacent if they are disjoint as subsets. A subset of $[m]$ is called $2$-stable if $2 \le |a-b| \le m-2$ for any distinct elements $a$ and $b$ in that subset. The reduced Kneser graph ${\sf KG}_2(m,n)$ is the subgraph of ${\sf KG}(m,n)$ induced by vertices that are $2$-stable subsets. We focus our study on the reduced Kneser graphs ${\sf KG}_2(2n+2,n)$. We achieve a complete analysis of its structure. From there, we derive that the circular chromatic number of ${\sf KG}_2(2n+2,n)$ is equal to its ordinary chromatic number, which is 4. A second application of the structural theorem shows that the chromatic index of ${\sf KG}_2(2n+2,n)$ is equal to its maximum degree except when $n=2$.