Colouring signed analogues of Kneser, Schrijver, and Borsuk graphs

Abstract

The Kneser signed graph $\KS(n,k)$, $k\le n$, is the graph whose vertices are signed $k$-subsets of $[n]$ (i.e. $k$-subsets $S$ of $\{\pm 1,\pm 2,\dots ,\pm n\}$ such that $S\cap (-S)=\varnothing$). Two vertices $A$ and $B$ are adjacent with a positive edge if $A\cap (-B)=\varnothing$ and with a negative edge if $A\cap B=\varnothing$. We prove that the balanced chromatic number of $\KS(n,k)$ is $n-k+1$. We then introduce the signed analogue of Schrijver graphs and show that they form vertex-critical subgraphs of $\KS(n,k)$ with respect to balanced colouring. Further connection to topological methods, in particular, connection to Borsuk signed graphs is also considered.