Bounded degree graphs and hypergraphs with no full rainbow matchings

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 contains exactly one edge from each color class $E_i$. One way to guarantee the existence 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 this paper, we apply a simple iterative method to construct edge-colored multi-hypergraphs with a given maximum degree, large color classes, and no full rainbow matchings. First, for every $r \ge 1$ and $\Delta \ge 2$, we construct edge-colored $r$-uniform multi-hypergraphs with maximum degree $\Delta$ such that each color class has size $|E_i| \ge r\Delta - 1$ and there is no full rainbow matching, which demonstrates that a theorem of Aharoni, Berger, and Meshulam (2005) is best possible. Second, we construct properly edge-colored multigraphs with no full rainbow matchings which disprove conjectures of Delcourt and Postle (2022). Finally, we apply results on full rainbow matchings to list edge-colorings and prove that a color degree generalization of Galvin's theorem (1995) does not hold

Rainbow matchings in properly colored multigraphs

Aharoni and Berger conjectured that in any bipartite multigraph that is properly edge-colored by $n$ colors with at least $n + 1$ edges of each color there must be a matching that uses each color exactly once. In this paper we consider the same question without the bipartiteness assumption. We show that in any multigraph with edge multiplicities $o(n)$ that is properly edge-colored by $n$ colors with at least $n + o(n)$ edges of each color there must be a matching of size $n-O(1)$ that uses each color at most once. Read More: https://epubs.siam.org/doi/abs/10.1137/17M115174