A d-graph G = (V ;E1, . . . , Ed) is a complete graph whose edges are colored by d colors, or in other words, are partitioned into d subsets (some of which might be empty). We say that G is complementary connected if the complement to each chromatic component of G is connected on V , or in other words, if for each two vertices u, w ∈ V and color i ∈ I = {1, . . . , d} there is a path between u ...

A bipartite graph with vertex sets X and Y is a triple {X, Y; E), where £ is a subset of X x Y, and a graph with vertex set X is a pair (X; £), where £ is a symmetric subset of X x X minus the diagonal. Nash-Williams [15] proposed the following generalization of these concepts. Let {X,$4,\x) and (Y, 08, v) be measure spaces. A bipartite measure graph with vertex sets X and Y is a triple (X, Y; ...

A d-graph G = (V ;E1, . . . , Ed) is a complete graph whose edges are colored by d colors, or in other words, are partitioned into d subsets (some of which might be empty). We say that G is complementary connected if the complement to each chromatic component of G is connected on V , or in other words, if for each two vertices u,w ∈ V and color i ∈ I = {1, . . . , d} there is a path between u a...

Our point of departure is the following simple common generalisation of the Sylvester-Gallai theorem and the Motzkin-Rabin theorem: Let S be a finite set of points in the plane, with each point coloured red or blue or with both colours. Suppose that for any two distinct points A, B ∈ S sharing a colour there is a third point C ∈ S, of the other colour, collinear with A and B. Then all the point...

As a generalization of matchings, Cunningham and Geelen introduced the notion of path-matchings. We give a structure theorem for path-matchings which generalizes the fundamental Gallai-Edmonds structure theorem for matchings. Our proof is purely combinatorial.