Generalised Mycielski graphs and bounds on chromatic numbers

TL;DR

This paper introduces a new approach using generalized Mycielski graphs to bound the chromatic number of graphs through topological and algebraic methods, providing polynomial-time solvable systems to estimate chromatic bounds.

Contribution

It develops a novel framework linking generalized Mycielski graphs with topological invariants and linear algebra to derive bounds on graph chromatic numbers.

Findings

Polynomial-time solvable linear systems for chromatic bounds

Conditions distinguishing between low and high chromatic numbers

Extension of methods to other algebraic bounds

Abstract

We prove that the coindex of the box complex $\mathrm{B}(H)$ of a graph $H$ can be measured by the generalised Mycielski graphs which admit a homomorphism to it. As a consequence, we exhibit for every graph $H$ a system of linear equations solvable in polynomial time, with the following properties: If the system has no solutions, then $\mathrm{coind}(\mathrm{B}(H)) + 2 \leq 3$; if the system has solutions, then $\chi(H) \geq 4$. We generalise the method to other bounds on chromatic numbers using linear algebra.