Properly colored Hamilton cycles in Dirac-type hypergraphs

TL;DR

This paper proves that in large k-uniform hypergraphs with high minimum codegree, any edge coloring satisfying certain local constraints guarantees the existence of properly colored Hamilton cycles, extending Dirac-type results.

Contribution

It introduces a robust variant of Dirac-type theorems for hypergraphs, establishing conditions for properly colored Hamilton cycles under edge colorings.

Findings

Properly colored tight Hamilton cycles exist under specified conditions.

Results extend to loose cycles in hypergraphs.

High minimum codegree ensures cycle existence with edge color constraints.

Abstract

We consider a robust variant of Dirac-type problems in $k$-uniform hypergraphs. For instance, we prove that if $H$ is a $k$-uniform hypergraph with minimum codegree at least $(1/2 + \gamma )n$, $\gamma >0$, and $n$ is sufficiently large, then any edge coloring $\phi$ satisfying appropriate local constraints yields a properly colored tight Hamilton cycle in $H$. Similar results for loose cycles are also shown.