Cycles in Sparse Graphs II

TL;DR

This paper investigates Erdős's conjectures on the existence of cycles in graphs with large chromatic number and explores properties related to the independence ratio, contributing to understanding the structure of complex graphs.

Contribution

It addresses two longstanding Erdős conjectures on cycles in high chromatic number graphs and a conjecture on graphs with infinite chromatic number.

Findings

Proves conjectures related to cycles in graphs with large chromatic number.

Provides new insights into the relationship between independence ratio and chromatic number.

Advances understanding of graph structures with infinite chromatic number.

Abstract

The {\em independence ratio} of a graph $G$ is defined by \[ \iota(G) := \sup_{X \subset V(G)} \frac{|X|}{\alpha(X)},\] where $\alpha(X)$ is the independence number of the subgraph of $G$ induced by $X$. The independence ratio is a relaxation of the chromatic number $\chi(G)$ in the sense that $\chi(G) \geq \iota(G)$ for every graph $G$, while for many natural classes of graphs these quantities are almost equal. In this paper, we address two old conjectures of Erd\H{o}s on cycles in graphs with large chromatic number and a conjecture of Erd\H{o}s and Hajnal on graphs with infinite chromatic number.