Partitioning the edge set of a hypergraph into almost regular cycles

Abstract

A cycle of length t in a hypergraph is an alternating sequence v1,e1,v2⋯,vt,et of distinct vertices vi and distinct edges ei so that false{vi,vi+1false}⊆ei (with vt+1:=v1). Let λKnh be the λ‐fold n‐vertex complete h‐graph. Let G=(V,E) be a hypergraph all of whose edges are of size at least h, and 2≤c1≤⋯≤ck≤false|Vfalse|. In order to partition the edge set of scriptG into cycles of specified lengths c1,⋯,ck, an obvious necessary condition is that 0true∑i=1kci=|E|. We show that this condition is sufficient in the following cases. (R1)h≥max{ck,false⌈n/2false⌉+1}. (R2)G=λKnh, h≥⌈n/2⌉+2. (R3)G=Knh, c1=⋯=ck:=c, c|n(n−1),n≥85.In (R2), we guarantee that each cycle is almost regular. In (R3), we also solve the case where a “small” subset L of edges of Knh is removed.

Large sets of Hamilton cycle decompositions of complete bipartite 3‐uniform hypergraphs

“…We will consider a different notion of Hamiltonicity which will be defined as follows, although there are others besides these, such as the loose Hamilton cycles defined by Kühn and Osthus . For the decomposition of hypergraphs with cycles, please see .…”