Rainbow factors in hypergraphs

Abstract

For any r-graph H, we consider the problem of finding a rainbow H-factor in an r-graph G with large minimum ℓ-degree and an edge-colouring that is suitably bounded. We show that the asymptotic degree threshold is the same as that for finding an H-factor. IntroductionA fundamental question in Extremal Combinatorics is to determine conditions on a hypergraph G that guarantee an embedded copy of some other hypergraph H. The Turán problem for an r-graph H asks for the maximum number of edges in an r-graph G on n vertices; we usually think of H as fixed and n as large. For r = 2 (ordinary graphs) this problem is fairly well understood (except when H is bipartite), but for general r and general H we do not even have an asymptotic understanding of the Turán problem (see the survey [11]). For example, a classical theorem of Mantel determines the maximum number of edges in a triangle-free graph on n vertices (it is n 2 /4 ), but we do not know even asymptotically the maximum number of edges in a tetrahedron-free 3-graph on n vertices. On the other hand, if we seek to embed a spanning hypergraph then it is most natural to consider minimum degree conditions. Such questions are known as Dirac-type problems, after the classical theorem of Dirac that any graph on n ≥ 3 vertices with minimum degree at least n/2 contains a Hamilton cycle. There is a large literature on such problems for graphs and hypergraphs, surveyed in [15,17,22,28].One of these problems, finding hypergraph factors, will be our topic for the remainder of this paper. To describe it we introduce some notation and terminology. Let G be an r-graph on [n] = {1, . . . , n}. For any L ⊆ V (G) the degree of L in G is the number of edges of G containing L. The minimum ℓ-degree δ ℓ (G) is the minimum degree in G over all L ⊆ V (G) of size ℓ. Let H be an r-graph with |V (H)| = h | n. A partial H-factor F in G of size m is a set of m vertex-disjoint copies of H in G. If m = n/h we call F an H-factor. We let δ ℓ (H, n) be the minimum δ such that δ ℓ (G) > δn r−ℓ guarantees an H-factor in G. Then the asymptotic ℓ-degree threshold for H-factors is δ * ℓ (H) := lim inf m→∞ δ ℓ (H, mh) .

Rainbow structures in locally bounded colorings of graphs

“…This topic has received considerable attention recently, with probabilistic tools and techniques from extremal graph theory allowing for major progress on longstanding problems. In this context, natural (rainbow) structures to seek include matchings, Hamilton cycles, spanning trees and triangle factors (see e.g., ). It is easy to see that results on edge‐colored also imply results on patterns in symmetric arrays.…”

Rainbow spanning structures in graph and hypergraph systems

“…, H m } and each H i is a properly colored k-graph. For general F -factors, Coulson et al [11] proved that essentially the minimum d-degree threshold guaranteeing an F -factor in a single k-graph also forces a rainbow F -factor in any edge-coloring of G that satisfies certain natural local conditions. We refer the reader to [35,1,4,8,14,38,31,34] for more results.…”

The n-queens completion problem

“…2 , we establish a connection between the n -queens completion problem and rainbow matchings in bipartite graphs. Then we deduce our lower bound modulo the proof of a “rainbow matching lemma.” The study of rainbow subgraph problems has been very fertile in recent years (see e.g., [ 1 , 3 , 8 – 10 , 13 , 15 , 17 , 18 , 22 – 25 , 30 ]). Our new rainbow matching lemma (Lemma 2.2 ) relates to this large body of recent results in Extremal Combinatorics.…”