Hamiltonian Chains in Hypergraphs

Gyula Y Katona

doi:10.46298/dmtcs.3444

Abstract

Hamiltionian chain is a generalisation of hamiltonian cycles for hypergraphs. Among the several possible ways of generalisations this is probably the most strong one, it requires the strongest structure. Since there are many interesting questions about hamiltonian cycles in graphs, we can try to answer these questions for hypergraphs, too. In the present article we give a survey on results about such questions.

Highlights

Let H be a r-uniform hypergraph on the vertex set V (H) = {v1, v2, . . . , vn} where n > r
An ordering (v1, v2, . . . , vl+1) of a subset of the vertex set is called an open chain of length l between v1 and vl+1 iff for each 1 ≤ i ≤ l − r + 2 there exists an edge Ej of H such that {vi, vi+1, . . . , vi+r−1} = Ej
A cyclic ordering (v1, v2, . . . , vl) of a subset of the vertex set is called a chain of length l iff for every 1 ≤ i ≤ l there exists an edge Ej of H such that {vi, vi+1, . . . , vi+r−1} = Ej

Summary

Introduction

Vl+1) of a subset of the vertex set is called an open chain of length l between v1 and vl+1 iff for each 1 ≤ i ≤ l − r + 2 there exists an edge Ej of H such that {vi, vi+1, . Vl) of a subset of the vertex set is called a chain of length l iff for every 1 ≤ i ≤ l there exists an edge Ej of H such that {vi, vi+1, . One reason of this is probably that the problem of deciding whether a given graph contains a hamiltonian cycle is an NP-complete problem From this point of view we cannot expect the hypergraph versions of these questions to be easier, since the corresponding problem for Hamiltionian chain is NP-complete for any fixed r. In each of the following four Sections we will deal with a separate question about hamiltonian chains

Dirac type theorem

Maximally non-hamiltonian hypergraphs

Hamiltonian chain saturated hypergraphs