Hamiltonicity and $\sigma$-hypergraphs

Christina Zarb

doi:10.20429/tag.2014.010101

Abstract

We define and study a special type of hypergraph. A σ-hypergraph H = H(n, r, q | σ), where σ 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 V1, V2,...,Vn, then a subset K of V (H) of size r is an edge if the partition of r formed by the non-zero cardinalities | K ∩ Vi |, 1 ≤ i ≤ n, is σ. The non-empty intersections K ∩ Vi are called the parts of K, and s(σ) 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 σ-hypergraphs contain a Hamiltonian Berge cycle and that, for n ≥ s + 1 and q ≥ r(r − 1), a σ-hypergraph H always contains a sharp Hamiltonian cycle. We also extend this result to k-intersecting cycles.

Highlights

IntroductionIn this paper we consider all the above types of Hamiltonian cycles in σ-hypergraphs
Let V = {v1, v2, ..., vn} be a finite set, and let E = {E1, E2, ..., Em} be a family of subsets of V
A σ-hypergraph H = H(n, r, q | σ), where σ is a partition of r, is an r-uniform hypergraph having nq vertices partitioned into n classes of q vertices each

Summary

Introduction

In this paper we consider all the above types of Hamiltonian cycles in σ-hypergraphs. When constructing sharp Hamiltonian cycles, we will use matchings — the link between matchings and Hamiltonian cycles in r-uniform hypergraphs has been extensively studied [1]. Given an r-uniform hypergraph H, a matching is a set of pairwise vertex-disjoint edges M ⊂ E(H). A perfect matching is a matching which covers all vertices of H and we denote the size of the largest matching in an r-uniform hypergraph H by ν(H). It is well known that in a graph, the Hamiltonian cycle yields a perfect or near perfect (leaving one vertex unmatched if n is odd) matching. We will show that these σ-hypergraphs, still have both a Berge and a sharp Hamiltonian cycle when q and n are large enough

Berge Cycles

Sharp Hamiltonian Cycles

Conclusion