Hamiltonicity and $\sigma$-hypergraphs

TL;DR

This paper introduces and analyzes a special class of hypergraphs called $\sigma$-hypergraphs, demonstrating that most contain Hamiltonian Berge cycles and, under certain conditions, sharp Hamiltonian cycles, extending to $k$-intersecting cycles.

Contribution

The paper defines $\sigma$-hypergraphs and proves they typically contain Hamiltonian Berge cycles and sharp Hamiltonian cycles under specific parameters.

Findings

Most $\sigma$-hypergraphs contain a Hamiltonian Berge cycle.

Under certain conditions, $\sigma$-hypergraphs always contain a sharp Hamiltonian cycle.

Results extend to $k$-intersecting cycles.

Abstract

We define and study a special type of hypergraph. A $\sigma$-hypergraph $H= H(n,r,q$ $\mid$ $\sigma$), where $\sigma$ is a partition of $r$, is an $r$-uniform hypergraph having $nq$ vertices partitioned into $ n$ classes of $q$ vertices each. If the classes are denoted by $V_1$, $V_2$,...,$V_n$, then a subset $K$ of $V(H)$ of size $r$ is an edge if the partition of $r$ formed by the non-zero cardinalities $ \mid$ $K$ $\cap$ $V_i \mid$, $ 1 \leq i \leq n$, is $\sigma$. The non-empty intersections $K$ $\cap$ $V_i$ are called the parts of $K$, and $s(\sigma)$ denotes the number of parts. We consider various types of cycles in hypergraphs such as Berge cycles and sharp cycles in which only consecutive edges have a nonempty intersection. We show that most $\sigma$-hypergraphs contain a Hamiltonian Berge cycle and that, for $n \geq s+1$ and $q \geq r(r-1)$, a $\sigma$-hypergraph $H$ always contains a sharp Hamiltonian cycle. We also extend this result to $k$-intersecting cycles.