Hamiltonian paths and cycles in some 4-uniform hypergraphs

2021/04/11 by Liu, Guanwu, Liu, Xiaonan
#05C07 #05C30 #05C35 #05C38 #05C45 #05C65 #Combinatorics (math.CO) #FOS: Mathematics

paper · doi:10.48550/arxiv.2104.05016

Abstract

In 1999, Katona and Kierstead conjectured that if a k-uniform hypergraph \cal H on n vertices has minimum co-degree \lfloor (n-k+3)/(2)\rfloor, i.e., each set of k-1 vertices is contained in at least \lfloor (n-k+3)/(2)\rfloor edges, then it has a Hamiltonian cycle. Rödl, Ruciński and Szemerédi in 2011 proved that the conjecture is true when k=3 and n is large. We show that this Katona-Kierstead conjecture holds if k=4, n is large, and V(\cal H) has a partition A, B such that |A|=\lceil n/2\rceil, |\e∈ E(\cal H):|e ∩ A|=2\| <εn⁴ for a fixed small constant ε>0.

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