DMS Discrete Mathematics Seminar

Speaker: Songling Shan (Auburn University)

Title: Toughness and the existence of \(k\)-factors in hypergraphs

Abstract: In 1973, Chvátal introduced the concept of graph toughness and initiated the study of hamiltonian cycles under toughness conditions. Many classic results related to graphs toughness were later on proved including the famous result by Enomoto,  Jackson, Keterinis, and Saito from 1985, that for every integer \(k\ge 1\), a graph \(G\) has a  \(k\)-factor if  \(G\) is \(k\)-tough, \(k |V(G)|\) is even, and \(|V(G)|\ge k+1\).

As a natural analogy of the concepts of cycles in graphs,  Berge cycles in hypergraphs have recently attracted more attentions of researchers and their existence was mostly studied under degree conditions.  In this paper, we study Berge \(k\)-factors, an analogy of \(k\)-factors,  in hypergraphs, and extend the classic result of Enomoto,  Jackson, Keterinis, and Saito to hypergraphs.  The proof is more complicated than the proof for the graph case as we do not require the hypergraph to be uniform.