A note on the chromatic number of Kneser graphs on chambers of projective planes and incidence-free sets

TL;DR

This paper provides an elementary proof of a perfect matching in incidence graphs of symmetric designs, linking the chromatic number of Kneser graphs on projective plane chambers to incidence-free numbers.

Contribution

It offers a new elementary proof for the existence of perfect matchings and connects the chromatic number problem to incidence-free sets in projective planes.

Findings

Elementary proof of perfect matching existence in incidence graphs

Equivalence between chromatic number of Kneser graphs and incidence-free numbers

Extended results to cases with k ≥ 36

Abstract

Let $D=(\mathcal{P},\mathcal{B})$ be a symmetric $(v,k,\lambda)$-design and let $(X,Y)$ be an equinumerous incidence-free pair, with $X\subseteq \mathcal{P}$ and $Y\subseteq \mathcal{B}$. In this note, we give an elementary proof which shows the existence of a perfect matching between $\mathcal{P} \setminus X$ and $\mathcal{B}\setminus Y$ in the incidence graph of $D$. This recovers a result of Spiro, Adriaensen and Mattheus, who already showed this using different arguments for $k\geq 36$. We use this to connect some dots in the literature and prove that finding the chromatic number of the Kneser graph on chambers of a projective plane is equivalent to finding the incidence-free number of the incidence graph of the plane.