Rainbow spanning structures in graph and hypergraph systems

Abstract: 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 $\textbf{G}={G_1, G_2,\ldots, G_{m}}$ of not necessarily distinct $k$-graphs on the same vertex set $[n]$, a (sub)graph $H$ on $[n]$ is rainbow if there exists an injection $\varphi: E(H)\rightarrow[m]$ such that $e\in E(G_{\varphi(e)})$ for each $e\in E(H)$. Note that if $|E(H)|=m$, then $\varphi$ is a bijection and thus $H$ contains exactly one edge from each $G_i$. 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--Szemer\'{e}di theorem, namely, if $t\mid n$ and $\delta(G_i)\geq(1-\frac{1}{t}+\varepsilon)n$ for each $i\in[\frac{n}{t}\binom{t}{2}]$, then $\textbf{G}$ contains a rainbow $K_t$-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 $d\in[k-1]$. 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 problems, which reduces it to subproblems with finitely many colors.