Colouring signed analogues of Kneser, Schrijver, and Borsuk graphs

TL;DR

This paper investigates the chromatic properties of signed analogues of classical graphs like Kneser, Schrijver, and Borsuk graphs, establishing their coloring numbers and topological connections.

Contribution

It introduces signed versions of Kneser and Schrijver graphs, determines their balanced chromatic number, and explores their topological and critical subgraph properties.

Findings

Balanced chromatic number of KS(n,k) is n-k+1.

Signed Schrijver graphs are vertex-critical subgraphs.

Connections to Borsuk signed graphs are established.

Abstract

The Kneser signed graph $\KS(n,k)$, $k\leq n$, is the graph whose vertices are signed $k$-subsets of $[n]$ (i.e. $k$-subsets $S$ of $\{ \pm 1, \pm 2, \ldots, \pm n\}$ such that $S\cap (-S)=\emptyset$). Two vertices $A$ and $B$ are adjacent with a positive edge if $A\cap (-B)=\emptyset$ and with a negative edge if $A\cap B=\emptyset$. 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.