Hamilton cycles in quasirandom hypergraphs

TL;DR

This paper proves that quasirandom hypergraphs with certain density and degree conditions contain loose Hamilton cycles, but also shows limitations for Hamilton $ ext{l}$-cycles depending on divisibility conditions, highlighting open problems.

Contribution

It establishes conditions under which quasirandom hypergraphs contain loose Hamilton cycles and identifies cases where such cycles do not exist, revealing new structural insights.

Findings

Quasirandom hypergraphs with specified density and degree contain loose Hamilton cycles.

Construction shows non-existence of Hamilton $ ext{l}$-cycles under certain divisibility conditions.

Open questions remain for cases where $k- ext{l}$ does not divide $k$.

Abstract

We show that, for a natural notion of quasirandomness in $k$-uniform hypergraphs, any quasirandom $k$-uniform hypergraph on $n$ vertices with constant edge density and minimum vertex degree $\Omega(n^{k-1})$ contains a loose Hamilton cycle. We also give a construction to show that a $k$-uniform hypergraph satisfying these conditions need not contain a Hamilton $\ell$-cycle if $k-\ell$ divides $k$. The remaining values of $\ell$ form an interesting open question.