Properly Colored Hamilton Cycles in Dirac-Type Hypergraphs

Abstract

We consider a robust variant of Dirac-type problems in $k$-uniform hypergraphs. For instance, we prove that if $\mathcal{H}$ is a $k$-uniform hypergraph with minimum codegree at least $\left(\frac 12 + \gamma \right)n$, $\gamma >0$, and $n$ is sufficiently large, then any edge coloring $\phi$ satisfying appropriate local constraints yields a properly colored tight Hamilton cycle in $\mathcal{H}$. Similar results for loose cycles are also shown.