Generalizing a result of Erdo s, Gya rfa s and Pyber we show that there exists a constant c such that for any integers r, k 2 and for any coloring of the edges of a complete graph with r colors, its vertices can be partitioned into at most r log r+k) connected monochromatic k-regular subgraphs and vertices. We also show that the same result holds for complete bipartite graphs, generalizing a re...

The mirror (or bipartite complement) mir(B) of a bipartite graph B = (X,Y,E) has the same color classes X and Y as B, and two vertices x ∈ X and y ∈ Y are adjacent in mir(B) if and only if xy / ∈ E. A bipartite graph is chordal bipartite if none of its induced subgraphs is a chordless cycle with at least six vertices. In this paper, we deal with chordal bipartite graphs whose mirror is chordal ...

Topological drawings are natural representations of graphs in the plane, where vertices are represented by points, and edges by curves connecting the points. Topological drawings of complete graphs and of complete bipartite graphs have been studied extensively in the context of crossing number problems. We consider a natural class of simple topological drawings of complete bipartite graphs, in ...

This survey paper deals with upper and lower bounds on the number of k-matchings in regular graphs on N vertices. For the upper bounds we recall the upper matching conjecture which is known to hold for perfect matchings. For the lower bounds we first survey the known results for bipartite graphs, and their continuous versions as the van der Waerden and Tverberg permanent conjectures and its var...

This survey paper deals with upper and lower bounds on the number of k-matchings in regular graphs on N vertices. For the upper bounds we recall the upper matching conjecture which is known to hold for perfect matchings. For the lower bounds we first survey the known results for bipartite graphs, and their continuous versions as the van der Waerden and Tverberg permanent conjectures and its var...

We consider bipartite subgraphs of sparse random graphs that are regular in the sense of Szemerédi and, among other things, show that they must satisfy a certain local pseudorandom property. This property and its consequences turn out to be useful when considering embedding problems in subgraphs of sparse random graphs.

With the help of a novel computational technique, we show that graphs with up to 11 vertices are determined uniquely by their sets of vertex-deleted subgraphs, even if the set of subgraphs is reduced by isomorphism type. The same result holds for triangle-free graphs to 14 vertices, square-free graphs to 15 vertices and bipartite graphs to 15 vertices, as well as some other classes.