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Given a multi-hypergraph $G$ that is edge-colored into color classes $E_1,\ldots, E_n$, a full rainbow matching is a matching of $G$ that containsexactly one edge from each color class $E_i$. One way to guarantee theexistence of a full rainbow matching is to have the size of each color class$E_i$ be sufficiently large compared to the maximum degree of $G$. In thispaper, we apply a simple iterative method to construct edge-coloredmulti-hypergraphs with a given maximum degree, large color classes, and no fullrainbow matchings. First, for every $r \ge 1$ and $\Delta \ge 2$, we constructedge-colored $r$-uniform multi-hypergraphs with maximum degree $\Delta$ suchthat each color class has size $|E_i| \ge r\Delta - 1$ and there is no fullrainbow matching, which demonstrates that a theorem of Aharoni, Berger, andMeshulam (2005) is best possible. Second, we construct properly edge-coloredmultigraphs with no full rainbow matchings which disprove conjectures ofDelcourt and Postle (2022). Finally, we apply results on full rainbow matchingsto list edge-colorings and prove that a color degree generalization of Galvin'stheorem (1995) does not hold.

Bounded degree graphs and hypergraphs with no full rainbow matchings