Kneser Graphs

Abstract

The Kneser graph K v:r is the graph with the r-subsets of a fixed v-set as its vertices, with two r-subsets adjacent if they are disjoint. We have already met the complete graphs K v:1, while K v:2 is the complement of the line graph of K v . The first half of this chapter is devoted to fractional versions of the chromatic number and clique number of a graph. We discover that for the fractional chromatic number, the Kneser graphs play a role analogous to that played by the complete graphs for the ordinary chromatic number. We use this setting to provide a proof of the Erdős-Ko-Rado theorem, which is a famous result from extremal set theory. In the remainder of the chapter, we determine the chromatic number of the Kneser graphs, which surprisingly uses a nontrivial result from topology, and study homomorphisms between Kneser graphs.