In the mathematical discipline of graph theory, a rainbow matching in an edge-colored graph is a matching in which all the edges have distinct colors.
Given an edge-colored graph, a rainbow matching in is a set of pairwise non-adjacent edges, that is, no two edges share a common vertex, such that all the edges in the set have distinct colors.
A maximum rainbow matching is a rainbow matching that contains the largest possible number of edges.
Rainbow matchings are of particular interest given their connection to transversals of Latin squares.
Denote by the complete bipartite graph on vertices. Every proper -edge coloring of corresponds to a Latin square of order . A rainbow matching then corresponds to a transversal of the Latin square, meaning a selection of positions, one in each row and each column, containing distinct entries.
This connection between transversals of Latin squares and rainbow matchings in has inspired additional interest in the study of rainbow matchings in triangle-free graphs.
An edge-coloring is called proper if each edge has a single color, and each two edges of the same color have no vertex in common.
A proper edge-coloring does not guarantee the existence of a perfect rainbow matching. For example, consider the graph : the complete bipartite graph on 2+2 vertices. Suppose the edges and are colored green, and the edges and are colored blue. This is a proper coloring, but there are only two perfect matchings, and each of them is colored by a single color. This invokes the question: when does a large rainbow matching is guaranteed to exist?
Much of the research on this question was published using the terminology of Latin transversals in Latin squares. Translated into the rainbow matching terminology:
A more general conjecture of Stein is that a rainbow matching of size exists not only for a proper edge-coloring, but for any coloring in which each color appears on exactly edges.
Some weaker versions of these conjectures have been proved:
Wang asked if there is a function such that every properly edge-colored graph with minimum degree and at least vertices must have a rainbow matching of size . Obviously at least vertices are necessary, but how many are sufficient?
Suppose that each edge may have several different colors, while each two edges of the same color must still have no vertex in common. In other words, each color is a matching. How many colors are needed in order to guarantee the existence of a rainbow matching?
Drisko[9] studied this question using the terminology of Latin rectangles. He proved that, for any, in the complete bipartite graph, any family of matchings (=colors) of size has a perfect rainbow matching (of size). He applied this theorem to questions about group actions and difference sets.
Drisko also showed that matchings may be necessary: consider a family of matchings, of which are and the other are Then the largest rainbow matching is of size (e.g. take one edge from each of the first matchings).
Alon[10] showed that Drisko's theorem implies an older result[11] in additive number theory.
Aharoni and Berger[12] generalized Drisko's theorem to any bipartite graph, namely: any family of matchings of size in a bipartite graph has a rainbow matching of size .
Aharoni, Kotlar and Ziv[13] showed that Drisko's extremal example is unique in any bipartite graph.
In general graphs, matchings are no longer sufficient. When is even, one can add to Drisko's example the matching and get a family of matchings without any rainbow matching.
Aharoni, Berger, Chudnovsky, Howard and Seymour[14] proved that, in a general graph, matchings (=colors) are always sufficient. It is not known whether this is tight: currently the best lower bound for even is and for odd it is .[15]
A fractional matching is a set of edges with a non-negative weight assigned to each edge, such that the sum of weights adjacent to each vertex is at most 1. The size of a fractional matching is the sum of weights of all edges. It is a generalization of a matching, and can be used to generalize both the colors and the rainbow matching:
It is known that, in a bipartite graph, the maximum fractional matching size equals the maximum matching size. Therefore, the theorem of Aharoni and Berger is equivalent to the following. Let be any positive integer. Given any family of fractional-matchings (=colors) of size in a bipartite graph, there exists a rainbow-fractional-matching of size .
Aharoni, Holzman and Jiang extend this theorem to arbitrary graphs as follows. Let be any positive integer or half-integer. Any family of fractional-matchings (=colors) of size at least in an arbitrary graph has a rainbow-fractional-matching of size . The is the smallest possible for fractional matchings in arbitrary graphs: the extremal case is constructed using an odd-length cycle.
For the case of perfect fractional matchings, both the above theorems can derived from the colorful Caratheodory theorem.
For every edge in, let be a vector of size, where for each vertex in, element in equals 1 if is adjacent to, and 0 otherwise (so each vector has 2 ones and -2 zeros). Every fractional matching corresponds to a conical combination of edges, in which each element is at most 1. A conical combination in which each element is exactly 1 corresponds to a perfect fractional matching. In other words, a collection of edges admits a perfect fractional matching, if and only if (the vector of ones) is contained in the conical hull of the vectors for in .
Consider a graph with vertices, and suppose there are subsets of edges, each of which admits a perfect fractional matching (of size). This means that the vector is in the conical hull of each of these subsets. By the colorful Caratheodory theorem, there exists a selection of edges, one from each subset, that their conical hull contains . This corresponds to a rainbow perfect fractional matching. The expression is the dimension of the vectors - each vector has elements.
Now, suppose that the graph is bipartite. In a bipartite graph, there is a constraint on the vectors : the sum of elements corresponding to each part of the graph must be 1. Therefore, the vectors live in a -dimensional space. Therefore, the same argument as above holds when there are only subsets of edges.
An r-uniform hypergraph is a set of hyperedges each of which contains exactly vertices (so a 2-uniform hypergraph is a just a graph without self-loops). Aharoni, Holzman and Jiang extend their theorem to such hypergraphs as follows. Let be any positive rational number. Any family of fractional-matchings (=colors) of size at least in an -uniform hypergraph has a rainbow-fractional-matching of size . The is the smallest possible when is an integer.
An r-partite hypergraph is an -uniform hypergraph in which the vertices are partitioned into disjoint sets and each hyperedge contains exactly one vertex of each set (so a 2-partite hypergraph is a just bipartite graph). Let be any positive integer. Any family of fractional-matchings (=colors) of size at least in an -partite hypergraph has a rainbow-fractional-matching of size . The is the smallest possible: the extremal case is when is a prime power, and all colors are edges of the truncated projective plane of order . So each color has edges and a fractional matching of size, but any fractional matching of that size requires all edges.[16]
For the case of perfect fractional matchings, both the above theorems can derived from the colorful caratheodory theorem in the previous section. For a general -uniform hypergraph (admitting a perfect matching of size), the vectors live in a -dimensional space. For an -uniform -partite hypergraph, the -partiteness constraints imply that the vectors live in a -dimensional space.
The above results hold only for rainbow fractional matchings. In contrast, the case of rainbow integral matchings in -uniform hypergraphs is much less understood. The number of required matchings for a rainbow matching of size grows at least exponentially with .
See also: matching in hypergraphs.
Garey and Johnson have shown that computing a maximum rainbow matching is NP-complete even for edge-colored bipartite graphs.[17]
Rainbow matchings have been applied for solving packing problems.[18]