Disjoint type graphs with no short odd cycles

TL;DR

This paper proves the existence of disjoint type graphs without short odd cycles, confirming a technical result by Erdős and Hajnal and demonstrating its optimality in a specific context.

Contribution

It provides a new proof of Erdős and Hajnal's result and shows that this result is sharp, advancing understanding of graph structures without short odd cycles.

Findings

Existence of disjoint type graphs with no short odd cycles

Proof of the sharpness of Erdős and Hajnal's result

Enhanced understanding of graph cycle constraints

Abstract

In this note, we provide a proof of a technical result of Erd\H{o}s and Hajnal about the existence of disjoint type graphs with no odd cycles. We also prove that this result is sharp in a certain sense.