In 1993 Voloshin introduced the concept of mixed hypergraph. A mixed hypergraph is characterised by the fact that it possesses anti-edges as well as edges. In a colouring of a mixed hypergraph, every anti-edge has at least two vertices of the same colour. In this paper a new parameter is introduced: the monochromatic block number mb. It is the number of monochromatic blocks in a colouring of a ...

We propose a natural cost function for the bi-clustering task, the monochromatic cost. This cost function is suitable for detecting meaningful homogeneous bi-clusters based on categorical valued input matrices. Such tasks arise in many applications, such as the analysis of social networks and in systemsbiology where researchers try to infer functional grouping of biological agents based on thei...

Let p1, p2,..., pn be a sequence of n pairwise coprime positive integers, and let P = p1p2...pn. Let 0, 1, ...,m − 1 be a sequence of m different colors. Let A be an n ×mP matrix of colors in which row i consists of blocks of pi consecutive entries of the same color, with colors 0 through m − 1 repeated cyclically. The Monochromatic Column problem is to determine the number of columns of A in w...

We conjecture that for n> 4(k−1) every 2-coloring of the edges of the complete graphKn contains a k-connectedmonochromatic subgraph with at least n − 2(k − 1) vertices. This conjecture, if true, is best possible. Here we prove it for k = 2, and show how to reduce it to the case n< 7k − 6. We prove the following result as well: for n> 16k every 2-colored Kn contains a k-connected monochromatic s...

We prove that any r-coloring of the edges of Km contains a monochromatic even cycle, where m = 3r + 1 if r is odd and m = 3r if r is even. We also prove that Km−1 has an r-coloring without monochromatic even cycles. An easy exercise, perhaps folkloristic, says that in any r-coloring of the edges of K2r+1 there is a monochromatic odd cycle (and this is not true for K2r). This note explores what ...

We consider a variation of a problem stated by Erdös and Guy in 1973 about the number of convex k-gons determined by any set S of n points in the plane. In our setting the points of S are colored and we say that a spanned polygon is monochromatic if all its points are colored with the same color. As a main result we show that any bi-colored set of n points in R in general position determines a ...