The Matching Kneser Graph Conjecture For High Chromatic Numbers

TL;DR

This paper constructs specific graphs demonstrating the relationships between clique number, chromatic number, and generalized chromatic parameters related to matching graphs, addressing conjectures in graph theory.

Contribution

It proves the existence of connected graphs with prescribed clique, chromatic, and matching-based parameters, advancing understanding of the Matching Kneser Graph Conjecture.

Findings

Existence of graphs with specified clique and chromatic numbers.

Construction of graphs with given matching chromatic parameters.

Insights into the structure of graphs related to the Matching Kneser Graph Conjecture.

Abstract

\noindent In this paper, we show that for any positive integers $r$, $k$, $\Theta$, and $\Gamma$ such that $k \geq 2$ and $r \geq k + \Gamma$, there exists a connected graph $G$ for which $$\begin{array}{llcr} \omega (G) = \chi (G) = k, & \chi \left( G , rK_2 \right) = \Theta , & {\rm and} & |E(G)| - {\rm ex}\left( G , rK_2 \right) = \Theta + \Gamma . \end{array}$$