Decomposing hypergraphs into cycle factors

Abstract

A famous result by Rödl, Ruciński, and Szemerédi guarantees a (tight) Hamilton cycle in k-uniform hypergraphs H on n vertices with minimum pk ´1q-degree δ k´1 pHq ě p1{2 `op1qqn, thereby extending Dirac's result from graphs to hypergraphs. For graphs, much more is known; each graph on n vertices with δpGq ě p1{2 `op1qqn contains p1 ´op1qqr edgedisjoint Hamilton cycles where r is the largest integer such that G contains a spanning 2r-regular subgraph, which is clearly asymptotically optimal. This was proved by Ferber, Krivelevich, and Sudakov answering a question raised by Kühn, Lapinskas, and Osthus.We extend this result to hypergraphs; every k-uniform hypergraph H on n vertices with δ k´1 pHq ě p1{2 `op1qqn contains p1 ´op1qqr edge-disjoint (tight) Hamilton cycles where r is the largest integer such that H contains a spanning subgraph with each vertex belonging to kr edges. In particular, this yields an asymptotic solution to a question of Glock, Kühn, and Osthus.In fact, our main result applies to approximately vertex-regular k-uniform hypergraphs with a weak quasirandom property and provides approximate decompositions into cycle factors without too short cycles.

Fractional cycle decompositions in hypergraphs

“…As we indicated earlier, Theorem 1.4 is also applied in a recent project of Schülke [7] and the authors concerning a generalization of the famous result in [12] where an analog of Dirac's theorem for hypergraphs is proved. We expect that Theorem 1.4 will be useful for further results in the area.…”

“…, k 2 )-connectedness. In another article together with Schülke [7], we employ Theorem 1.4 to prove a strong generalization of the well-known result due to Rödl, Ruciński, and Szemerédi [12]. We show that every k-graph H on n vertices with minimum degree 𝛿(H) ≥ (1∕2 + o(1))n not only contains one Hamilton cycle but essentially as many edge-disjoint Hamilton cycles as H may potentially have, namely, the largest p for which H has a spanning subgraph where every vertex is contained in kp edges.…”