Work

Digraphs of Large Girth and Dichromatic Number in Tournaments with Large Dichromatic Number

Year: 2026

Type: preprint

Abstract: In the 1960s, Erdős and Hajnal conjectured that every graph with sufficiently large chromatic number contains a subgraph of large girth (size of a smallest cycle) and large chromatic number. In this paper, we prove that every tournament with sufficiently large dichromatic number contains a subdigraph of large digirth (size of a smallest directed cycle) and large dichromatic number. We investigate the same statement when replacing digirth by girth (of the underlying graph). We show that it implies the conjecture of Erdős and Hajnal, and prove it for a particular family of tournaments.

Source: arXiv (Cornell University)

DOI: 10.48550/arxiv.2609.21895

Authors: Charbit, Pierre, Coulomb, Samuel

Institutions: Centre National de la Recherche Scientifique, Université Paris Cité, Institut de Recherche en Informatique Fondamentale

Language: English

Topic: Limits and Structures in Graph Theory

Subfield: Discrete Mathematics and Combinatorics

Field: Mathematics

Domain: Physical Sciences

Open Access status: green

OpenAlex ID: W7213896377

Cites:

Cited by:

Related to: 10