Transversal factors and spanning trees

Abstract

Since the proof of a "colorful" version of [Caratheodory's theorem](https://en.wikipedia.org/wiki/Carath%C3%A9odory%27s_theorem_%28convex_hull%29) by Bárány in 1982, it has been an important problem to obtain colorful extensions of other classical results in discrete geometry (for instance Tverberg's theorem). The present paper continues this line of research, but in the context of extremal graph theory rather than discrete geometry. Mantel's classical theorem from 1907 states that every $n$-vertex graph on more than $n^2/4$ edges contains a triangle. In [Ron Aharoni, Matt DeVos, Sebastián González Hermosillo de la Maza, Amanda Montejano, and Robert Šámal, A rainbow version of Mantel’s Theorem, Advances in Combinatorics 2020:2, 12 pp](https://arxiv.org/abs/1812.11872v2), a "rainbow", "colored", or "colorful" variant of this problem was considered : given three graphs $G_1,G_2,G_3$ on the same vertex set of size $n$, what average degree conditions on $G_1,G_2,G_3$ force the existence of a "rainbow triangle" (a triangle $\{e_1,e_2,e_3\}$ such that each edge $e_i$ belongs to $G_i$)? By taking three copies of the same graph $G$ we see that the colored version is at least as hard as the original problem, and the paper cited above provided a construction showing that in this case the colorful variant is strictly harder than Mantel's problem. It was suggested to study average degree or minimum degree thresholds for colorful variants of classical problems in extremal combinatorics, such as Dirac's theorem (every $n$-vertex graph of minimum degree at least $n/2$ has a Hamiltonian cycle). In particular, Joos and Kim proved in 2020 that the same minimum degree condition as in Dirac's theorem guarantees a rainbow $n$-cycle: namely if we are given $n$ graphs of minimum degree at least $n/2$ on the same set of $n$ vertices, then there is an $n$-cycle comprising one edge of each graph. The results in the present paper follow the same line of research. The two major results that are extended to the colorful setting here are a theorem of Kühn and Osthus (a sharp minimum degree condition to obtain a perfect packing of copies of any given graph $F$, generalizing the Hajnal-Szemerédi theorem), and a theorem of Komlós, Sárközy and Szemerédi (a sharp degree condition to contain any given spanning tree without large degree vertices). Amazingly, the minimum degree conditions in the (stronger) colorful versions are the same as the original minimum degree conditions.

Adaptive hypergraph superpixels

“…Inspired by a series of very recent successes on rainbow matchings [29,28,30,31], rainbow Hamilton cycles [8,9,21] and rainbow factors [7,12,33], we suspect the threshold for a rainbow spanning subgraph in (hyper)graph system is asymptotically same with the threshold for a spanning subgraph in a (hyper)graph.…”

A bandwidth theorem for graph transversals

“…Note that this lemma is not stated exactly as above in [30]. For completeness, we deduce it from the lemma below stated in [30]. Proof of Lemma 2.21.…”

“…[h] the η-fraction graph of the graph system G. This concept of fraction graph was introduced in [30] by Montgomery, Müyesser and Pehova. They observed that this concept is very useful for finding graph partial transversals.…”

“…For the proof of Theorem 1.2, we develop the ideas from [30], where Montgomery, Müyesser and Pehova laid out a general strategy to find a G-transversal isomorphic to a spanning graph H in the collection G = (G i : i ∈ [h]). Using low bandwidth of H, we first partition H into vertex-disjoint induced subgraphs H 0 , .…”

“…2 contains a G-transversal isomorphic to a Hamilton cycle. Montgomery, Müyesser and Pehova [30] determined asymptotically tight bound on δ(G) for the existence of a G-transversal isomorphic to a general F -factor or a spanning tree with maximum degree at most o( n log n ), generalizing the classical Hajnal-Szemerédi theorem [19] and the tree-embedding theorem of Komlós, Sárközy and Szmerédi [27] respectively. A similar result for K tfactor was proved by Cheng, Han, Wang and Wang [10] and generalized into the cases of digraphs and hypergraphs, and Cheng, Wang and Zhao [12] further proved results for Hamilton cycles in the collections of hypergraphs.…”