Systolic inequalities and chromatic number

TL;DR

This paper explores how discrete systolic inequalities can be applied to graphs, providing improved estimates for the number of vertices based on chromatic number and odd cycle length, with enhanced bounds using graph-theoretic methods.

Contribution

It introduces a novel application of discrete systolic inequalities to graph theory, improving vertex estimates related to chromatic number and odd cycles.

Findings

Systolic inequalities give lower bounds on graph vertices based on chromatic number.

Enhanced estimates achieved by combining systolic methods with graph-theoretic techniques.

Applications lead to better understanding of graph structure and properties.

Abstract

We show that the discrete versions of the systolic inequality that estimate the number of vertices of a simplicial complex from below have substantial applications to graphs, the one-dimensional simplicial complexes. Almost directly they provide good estimates for the number of vertices of a graph in terms of its chromatic number and the length of the smallest odd cycle. Combined with the graph-theoretic techniques of Berlov and Bogdanov, the systolic approach produces even better estimates.