In this paper, we establish a couple of results on extremal problems in bipartite graphs. Firstly, show that every sufficiently large graph with average degree $D$ and $n$ vertices each side has balanced independent set containing $(1-\epsilon) \frac{\log D}{D} n$ from for small $\epsilon > 0$. Secondly, prove the vertex maximum at most $\Delta$ can be partitioned into $(1+\epsilon)\frac{\Delta...

We establish several geometric extensions of the Lipton-Tarjan separator theorem for planar graphs. For instance, we show that any collection C of Jordan curves in the plane with a total of m crossings has a partition into three parts C = S ∪C1 ∪C2 such that |S| = O( √ m), max{|C1|, |C2|} ≤ 2 3 |C|, and no element of C1 has a point in common with any element of C2. These results are used to obt...

We consider a distance-regular graph ? with diameter d 3 and eigenvalues k = 0 > 1 > > d. We show the intersection numbers a 1 ; b 1 satisfy (a 1 + 1) 2 : We say ? is tight whenever ? is not bipartite, and equality holds above. We characterize the tight property in a number of ways. For example, we show ? is tight if and only if the intersection numbers are given by certain rational expressions...

Edge-intersection graphs of paths in grids are graphs that can be represented with vertices as paths in grids and edges between the vertices of the graph exist whenever two grid paths share a grid edge. This type of graphs is motivated by applications in conflict resolution of paths in grid networks. In this paper, we continue the study of edge-intersection graphs of paths in a grid, which was ...

Basic path-matchings, introduced by Cunningham and Geelen (FOCS 1996), are a common generalization of matroid intersection and non-bipartite matching. The main results of this paper are a new algebraic characterization of basic path-matching problems and an algorithm for constructing basic path-matchings in Õ(nω) time, where n is the number of vertices and ω is the exponent for matrix multiplic...

For a graph G , we define the lower bipartite number LB( ) as minimum order of maximal induced subgraph . We study parameter, and related parameter domination, providing bounds both in general graphs some families. example, show that there are arbitrarily large 4-connected planar with = 4 but 5-connected has linear ). also if is outerplanar n then lies between ( + 2)/3 2 /3, these sharp.