Rainbow spanning structures in graph and hypergraph systems

We study the following rainbow version of subgraph containment problems in a family of (hyper)graphs, which generalizes the classical subgraph containment problems in a single host graph. For a collection of not necessarily distinct k-graphs on the same vertex set, a (sub)graph H on is rainbow if there exists an injection, such that for each. Note that if, then is a bijection, and thus H contains exactly one edge from each. Our main results focus on rainbow clique-factors in (hyper)graph systems with minimum d-degree conditions. Specifically, we establish the following: (1) A rainbow analogue of an asymptotical version of the Hajnal-Szemerédi theorem, namely, if and for each, then contains a rainbow -factor; (2) Essentially, a minimum d-degree condition forcing a perfect matching in a k-graph also forces rainbow perfect matchings in k-graph systems for. The degree assumptions in both results are asymptotically best possible (although the minimum d-degree condition forcing a perfect matching in a k-graph is in general unknown). For (1), we also discuss two directed versions and a multipartite version. Finally, to establish these results, we in fact provide a general framework to attack this type of problem, which reduces it to subproblems with finitely many colors.