Minimum vertex degree conditions for loose Hamilton cycles in 3-uniform hypergraphs

Abstract

Abstract. We investigate minimum vertex degree conditions for 3-uniform hypergraphs which ensure the existence of loose Hamilton cycles. A loose Hamilton cycle is a spanning cycle in which only consecutive edges intersect and these intersections consist of precisely one vertex.We prove that every 3-uniform n-vertex (n even) hypergraph H with minimum vertex degree δ 1 pHq ě`7 16`o p1q˘`n 2˘c ontains a loose Hamilton cycle. This bound is asymptotically best possible. §1. IntroductionWe consider k-uniform hypergraphs H " pV, Eq with vertex sets V " V pHq and edge sets E " EpHq Ď`V k˘, where`V k˘d enotes the family of all k-element subsets of the set V . We often identify a hypergraph H with its edge set, i.e., H Ď`V k˘, and for an edge tv 1 , . . . , v k u P H we often suppress the enclosing braces and write v 1 . . . v k P H instead. Given a k-uniform hypergraph H " pV, Eq and a set S " tv 1 , . . . , v s u P`V sl et degpSq " degpv 1 , . . . , v s q denote the number of edges of H containing the set S and let N pSq " N pv 1 , . . . , v s q denote the set of those pk´sq-element sets T P`V k´s˘s uch that S Y T forms an edge in H. We denote by δ s pHq the minimum s-degree of H, i.e., the minimum of degpSq over all s-element sets S Ď V . For s " 1 the corresponding minimum degree δ 1 pHq is referred to as minimum vertex degree whereas for s " k´1 we call the corresponding minimum degree δ k´1 pHq the minimum collective degree of H.We study sufficient minimum degree conditions which enforce the existence of spanning, so-called Hamilton cycles. A k-uniform hypergraph C is called an -cycle if there is a cyclic ordering of the vertices of C such that every edge consists of k consecutive vertices, every vertex is contained in an edge and two consecutive edges (where the ordering of the edges 2010 Mathematics Subject Classification. 05C65 (primary), 05C45 (secondary).

Minimum vertex degree condition for tight Hamiltonian cycles in 3‐uniform hypergraphs

“…Given a hypergraph , a cycle, tight or loose, is called Hamiltonian if it is a subhypergraph of passing through all the vertices of . The optimal approximate minimum pair and vertex degree conditions for the existence of loose Hamiltonian cycles were obtained in and precise versions for large hypergraphs appeared in . Results on pair degree conditions implying tight Hamiltonian cycles were obtained in .…”

A pair degree condition for Hamiltonian cycles in $3$-uniform hypergraphs

“…Let α, L, ϑ be as in the statement. Choose n P N such that we have the hierarchy 1 " α, 1 L " ϑ " 1 n . Let H, R, R 1 be as in the statement of the Lemma.…”

“…One can also consider degree conditions for loose Hamiltonian cycles, looking at loose paths and cycles in which consecutive edges only intersect in one vertex. Loose Hamiltonian cycles were for instance studied in [1,4,7,10].…”

“…Rödl, Ruciński, and Szemerédi [15] started by showing that for given α ą 0, there is n 0 P N such that every 3-graph on n ě n 0 vertices with minimum pair-degree at least p 1 2 `αqn contains a Hamiltonian cycle. Actually, in [16] they improved the result to the following, which is optimal (as minimum pair-degree condition): Theorem 1.2.…”

Loose Hamiltonian cycles forced by large (k− 2)-degree - sharp version

“…In [9], Kühn, Mycroft, and Osthus generalized this result to 1 ď ℓ ă k, settling the problem of the existence of Hamiltonian ℓ-cycles in k-uniform hypergraphs with large minimum pk´1q-degree. In Theorem 1 below (see [1,2]) we have minimum pk´2q-degree conditions that ensure the existence of Hamiltonian ℓ-cycles for 1 ď ℓ ă k{2. Theorem 1.…”