Let G be a graph and H be a hypergraph both on the same vertex set. We say that a hypergraph H is a Berge-G if there is a bijection f : E(G) → E(H) such that for e ∈ E(G) we have e ⊂ f(e). This generalizes the established definitions of “Berge path” and “Berge cycle” to general graphs. For a fixed graph G we examine the maximum possible size (i.e. the sum of the cardinality of each edge) of a h...

Throughout this paper only undirected simple graphs (without loops or multiple edges) are considered. Unless stated otherwise, we follow the book by Bollobás [2] for undefined terminology and definitions. LetG = G(m,n) = G(X,Y ) denote a bipartite graph with vertex classesX and Y such that |X| = m and |Y | = n. A complete bipartite subgraph with s vertices in the class X and t vertices in the c...

Zarankiewicz’s Conjecture (ZC) states that the crossing number cr(Km,n) equals Z(m,n) := bm2 cb m−1 2 cb n 2 cb n−1 2 c. Since Kleitman’s verification of ZC for K5,n (from which ZC for K6,n easily follows), very little progress has been made around ZC; the most notable exceptions involve computer-aided results. With the aim of gaining a more profound understanding of this notoriously difficult ...

We show tight necessary and sufficient conditions on the sizes of small bipartite graphs whose union is a larger bipartite graph that has no large bipartite independent set. Our main result is a common generalization of two classical results in graph theory: the theorem of Kővári, Sós and Turán on the minimum number of edges in a bipartite graph that has no large independent set, and the theore...

The well-known Zarankiewicz problem [Za] is to determine the least positive integer Z(m,n, r, s) such that each m × n 0-1 matrix containing Z(m,n, r, s) ones has an r × s submatrix consisting entirely of ones. In graph-theoretic language, this is equivalent to finding the least positive integer Z(m, n, r, s) such that each bipartite graph on m black vertices and n white vertices with Z(m,n, r, ...