Rainbow Matchings: existence and counting

A perfect matching M in an edge-colored complete bipartite graph K_{n,n} is rainbow if no pair of edges in M have the same color. We obtain asymptotic enumeration results for the number of rainbow matchings in terms of the maximum number of occurrences of a color. We also consider two natural models of random edge-colored K_{n,n} and show that, if the number of colors is at least n, then there is with high probability a random matching. This in particular shows that almost every square matrix of order n in which every entry appears at most n times has a Latin transversal.

💡 Research Summary

The paper investigates two fundamental questions concerning rainbow perfect matchings in edge‑colored complete bipartite graphs K_{n,n}. A rainbow matching is a set of n pairwise disjoint edges, one incident to each vertex on both sides, such that no two edges share the same colour. The authors focus first on deterministic colourings and then on random colourings, deriving asymptotic enumeration formulas and high‑probability existence results, respectively.

In the deterministic setting the key parameter is m, the maximum number of times any single colour appears among the n² edges. The main enumeration theorem shows that when m=o(n) the number of rainbow matchings is ((1‑o(1))·n!). The proof combines a switching argument with inclusion‑exclusion: “bad” colours that appear m times are treated as obstacles, and the number of matchings avoiding them is bounded from below, while an upper bound follows from classical permanent estimates. When m is a linear fraction of n, the authors obtain matching upper and lower bounds up to a constant factor, demonstrating that the dependence on m is essentially tight.

The second part studies two natural random colour models. Model A colours each edge independently and uniformly from a palette of c≥n colours. Model B first fixes a palette of c≥n colours and then assigns exactly n edges to each colour uniformly at random (so each colour appears exactly n times). For both models the authors compute the first and second moments of the random variable counting rainbow matchings. In Model A, the expected number of rainbow matchings equals (n!·(c/n)^n); concentration is obtained via Chebyshev’s inequality, yielding a probability of existence (1‑o(1)). In Model B the colour frequencies are perfectly balanced, which eliminates heavy tails; using Chernoff‑type bounds the authors show that the probability a given colour blocks a matching decays exponentially. Consequently, with high probability a rainbow perfect matching exists in both models whenever the number of colours is at least n.

These probabilistic existence results have a direct combinatorial corollary. An n×n matrix whose entries are drawn from a set of symbols, each symbol occurring at most n times, can be viewed as a colour‑assignment in Model B. The high‑probability existence of a rainbow matching translates into the statement that “almost every” such matrix possesses a Latin transversal (a selection of n entries, one from each row and column, all distinct). This strengthens earlier results that required each symbol to appear at most once per row or column.

The paper concludes with a discussion of related work on rainbow matchings, permanents, and Latin transversals, and it outlines several open problems. Notably, determining the exact constant factor in the enumeration when m=Θ(n) remains unresolved, as does extending the methods to non‑complete bipartite graphs or to colourings where the colour distribution is highly skewed. Overall, the work provides a unified framework that bridges deterministic combinatorial enumeration with probabilistic existence theory, offering new insight into classical problems such as the existence of Latin transversals in dense matrices.