We prove a geometric version of the graph separator theorem for the unit disk intersection graph: for any set of n unit disks in the plane there exists a line ` such that ` intersects at most O( √ (m+ n) logn) disks and each of the halfplanes determined by ` contains at most 2n/3 unit disks from the set, where m is the number of intersecting pairs of disks. We also show that an axis-parallel li...

In a distance-regular graph, the partition with respect to distance from a vertex supports a unique eigenvector for each eigenvalue. There may be non-singleton vertex sets whose corresponding distance partition also supports eigenvectors. We consider the members of three families of distance regular graphs, the Johnson Graphs, Hamming Graphs and Complete Multipartite graphs. For each we determi...

We study balanced circle packings and circle-contact representations for planar graphs, where the ratio of the largest circle’s diameter to the smallest circle’s diameter is polynomial in the number of circles. We provide a number of positive and negative results for the existence of such balanced configurations.

We introduce and study balanced online graph avoidance games on the random graph process. The game is played by a player we call Painter. Edges of the complete graph with n vertices are revealed two at a time in a random order. In each move, Painter immediately and irrevocably decides on a balanced coloring of the new edge pair: either the first edge is colored red and the second one blue or vi...

For a graph G = (V,E) of even order, a partition (V1, V2) of the vertices is said to be perfectly balanced if |V1| = |V2| and the numbers of edges in the subgraphs induced by V1 and V2 are equal. For a base graph H define a random graph G(H, p) by turning every non-edge of H into an edge and every edge of H into a nonedge independently with probability p. We show that for any constant ǫ there i...

A random geometric graph G(n, r) is obtained by spreading n points uniformly at random in a unit square, and by associating a vertex to each point and an edge to each pair of points at Euclidian distance at most r. Such graphs are extensively used to model wireless ad-hoc networks, and in particular sensor networks. It is well known that, over a critical value of r, the graph is connected with ...

An oriented hypergraph is an oriented incidence structure that extends the concept of a signed graph. We introduce hypergraphic structures and techniques central to the extension of the circuit classification of signed graphs to oriented hypergraphs. Oriented hypergraphs are further decomposed into three families – balanced, balanceable, and unbalanceable – and we obtain a complete classificati...

Consider the following one-player game. Starting with the empty graph on n vertices, in every step r new edges are drawn uniformly at random and inserted into the current graph. These edges have to be colored immediately with r available colors, subject to the restriction that each color is used for exactly one of these edges. The player’s goal is to avoid creating a monochromatic copy of some ...

We show that deciding if a mixed graph has a well-balanced orientation is NP -complete.

We prove that if every subgraph of a graph G has a balanced separation of order at most a then G has treewidth at most 105a. This establishes a linear dependence between the treewidth and the separation number.