Bounded degree graphs and hypergraphs with no full rainbow matchings

TL;DR

This work studies full rainbow matchings in edge-colored multi-hypergraphs, showing that large color classes relative to the maximum degree do not guarantee existence of a full rainbow matching. It develops an iterative join-lemma construction to build $r$-graphs with maximum degree $oldsymbol{ riangle}$ and color classes of size at least $roldsymbol{ riangle}-1$ that lack such matchings, establishing the optimality of the Aharoni–Berger–Meshulam bound. Extending to properly edge-colored multigraphs, the paper disproves Delcourt–Postle conjectures and links rainbow matchings to list-edge-coloring, proving that a color-degree generalization of Galvin’s theorem fails. The results rely on nets, sunflowers, and blow-up techniques, and yield insights with implications for list-edge-coloring, chromatic indices, and related extremal problems in hypergraphs and graphs.

Abstract

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 $Δ\ge 2$, we construct edge-colored $r$-uniform multi-hypergraphs with maximum degree $Δ$ such that each color class has size $|E_i| \ge rΔ- 1$ 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.

Figures (4)

• Figure 1: An illustration of iteratively applying Lemma \ref{['join-lemma']} to the $4$-cycle $C_4$ with a proper $2$-edge-coloring.
• Figure 2: The disjoint union of five copies of the $(3,2)$-net, i.e. the $3 \times 3$ grid, edge-colored into color classes each of size $3 \cdot 2 - 1 = 5$ with no full rainbow matching.
• Figure 3: For maximum degree $\Delta = 4$, the list edge-cover graph $G_0$ and its edge-color classes $\mathcal{P}_0$ with no full rainbow matching. One color class has size 1, while the other color classes have size $\Delta$.
• Figure 4: For maximum color degree $\Delta = 4$, the bipartite graph $H_0$ and list assignment $L_0$ for $E(H_0)$ with no proper $L_0$-coloring. One list has size $1$, while the other lists have size $\Delta$. The colors in the list $L_0(e_i)$ correspond to the connected components in Figure \ref{['list-edge-cover']} that contain an edge with color $i$.

Theorems & Definitions (38)

• Theorem 1: Aharoni, Berger, Meshulam
• Theorem 2
• Theorem 3
• Theorem 4: Delcourt, Postle
• Conjecture 5: Delcourt, Postle
• Conjecture 6: Delcourt, Postle
• Theorem 7
• Theorem 8: Galvin
• Theorem 9
• Theorem 10