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A rainbow Dirac theorem for loose Hamilton cycles in hypergraphs

Part of: Extremal combinatorics Graph theory

Published online by Cambridge University Press:  11 November 2025

Amarja Kathapurkar ,
Patrick Morris  and
Guillem Perarnau
Amarja Kathapurkar
Affiliation:
University of Birmingham, Birmingham, UK
Patrick Morris*
Affiliation:
Universitat Politècnica de Catalunya (UPC), Barcelona, Spain
Guillem Perarnau
Affiliation:
Universitat Politècnica de Catalunya (UPC), Barcelona, Spain Centre de Recerca Matemàtica, Bellaterra, Spain
*
Corresponding author: Patrick Morris; Email: pmorrismaths@gmail.com
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Abstract

A meta-conjecture of Coulson, Keevash, Perarnau, and Yepremyan [12] states that above the extremal threshold for a given spanning structure in a (hyper-)graph, one can find a rainbow version of that spanning structure in any suitably bounded colouring of the host (hyper-)graph. We solve one of the most pertinent outstanding cases of this conjecture by showing that for any $1\leq j\leq k-1$, if $G$ is a $k$-uniform hypergraph above the $j$-degree threshold for a loose Hamilton cycle, then any globally bounded colouring of $G$ contains a rainbow loose Hamilton cycle.

Keywords

Spanning structuresHypergraphsextremal graph theorylocal lemma

MSC classification

Primary: 05C35: Extremal problems 05D40: Probabilistic methods
Secondary: 05C65: Hypergraphs

Information

Type
Paper
Information
Combinatorics, Probability and Computing , First View , pp. 1 - 29
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Copyright
© The Author(s), 2025. Published by Cambridge University Press

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Footnotes

*

Research supported by EPSRC Research grant EP/R034389/1.

†

Research supported by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) Walter Benjamin programme - project number 504502205 and by the European Union’s Horizon Europe Marie Sk lodowska-Curie grant RAND-COMB-DESIGN - project number 101106032 .

‡

Research supported by the grants RED2022-134947-T, PID2023-147202NB-I00, PCI2024-155080-2 and the Programme Severo Ochoa y María de Maeztu por Centros y Unidades de Excelencia en I&D (CEX2020-001084-M), all of them funded by MICIU/AEI/10.13039/501100011033.

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