Understanding evolutionary dynamics from a systemic point of view crucially depends on knowledge about how evolution affects size and structure of the organisms' functional building blocks (modules). It has been recently reported that statistics over sparse PPI graphlets can robustly monitor such evolutionary changes. However, there is abundant evidence that in PPI networks modules can be ident...

In the past we have published quite a few papers on chromatic numbers of graphs (finite or infinite), we give a list of those which are relevant to our present subject in the references . In this paper we will mainly deal with problems of the following type : Assuming that the chromatic number x(1) of a graph is greater than K, a finite or infinite cardinal, what can be said about the behaviour...

The topic of this paper is the rectangle-free coloring of grids using four colors which is equivalent to the edge coloring of complete bipartite graphs without complete monochromatic subgraphs K2,2. So far unsolved are the grids of the sizes 17×17, 17×18, 18×17, and 18×18. The number of different 4-color patterns of the grid 18×18 is equal to 4324 ≈ 1.16798∗10195 . We summarize in this paper so...

A = {(1, 0), (2, 0), . . . , (n, 0)}, B = {((1, 1), (2, 1), . . . , (n, 1)} and the edge ab is the line segment joining a ∈ A and b ∈ B in R. This model is essentially the same as the cyclic bipartite graphs and ordered bipartite graphs considered earlier by several authors. Subgraphs — paths, trees, double stars, matchings — are called non-crossing if they do not contain edges with common inte...

Many real world applications rely on the discovery of maximal biclique subgraphs (complete bipartite subgraphs). However, existing algorithms for enumerating maximal bicliques are not very efficient in practice. In this paper, we propose an efficient algorithm to mine large maximal biclique subgraphs from undirected graphs. Our algorithm uses a divide-and-conquer approach. It effectively uses t...

A set of vertices $S$ in a connected graph $G$ is a different-distance set if, for any vertex $w$ outside $S$, no two vertices in $S$ have the same distance to $w$.The lower and upper different-distance number of a graph are the order of a smallest, respectively largest, maximal different-distance set.We prove that a different-distance set induces either a special type of path or an independent...