We consider the typical distance between vertices of the giant component of a random intersection graph having a power law (asymptotic) vertex degree distribution with infinite second moment. Given two vertices from the giant component we construct OP (log log n) upper bound for the length of the shortest path connecting them. key words: intersection graph, random graph, power law, giant compon...

The concept of intersection numbers of order r for t-designs is generalized to graphs and to block designs which are not necessarily t-designs. These intersection numbers satisfy certain integer linear equations involving binomial coefficients, and information on the nonnegative integer solutions to these equations can be obtained using the block intersection polynomials introduced by P.J. Came...

Constructions and bounds are given for the largest clique that can be represented as an intersection graph of intervals on the real line with bounded depth and multiplicity. Also, for the graph defined on the subsets of a set by putting vertices adjacent if and only if the corresponding subsets intersect, the interval number and boxicity are examined. Ke ywords: intersection graph, interval num...

Let T be a (first order complete) dependent theory, C a κ̄-saturated model of T and G a definable subgroup which is abelian. Among subgroups of bounded index which are the union of < κ̄ type definable subsets there is a minimal one, i.e. their intersection has bounded index. See history in [Sh:876]. We then deal with definable groups for 2-dependent theories, a wider class of first order theories...

A unit disk graph is the intersection graph of unit disks in the euclidean plane. We present a polynomial-time approximation scheme for the maximum weight independent set problem in unit disk graphs. In contrast to previously known approximation schemes, our approach does not require a geometric representation (specifying the coordinates of the disk centers). The approximation algorithm present...

A Steiner triple system of order n is a collection of subsets of size three, taken from the n-element set {0, 1, ..., n−1}, such that every pair is contained in exactly one of the subsets. The subsets are called triples, and a block-intersection graph is constructed by having each triple correspond to a vertex. If two triples have a non-empty intersection, an edge is inserted between their vert...

In this paper, we show that many graph search problems can be solved quite efficiently for a geometric intersection graph ofhorizontal and vertical line segments. We first extract several basic operations for depth first search and breadth first search on a graph. Then we present data structures for the intersection graph in terms of which those operations can be implemented in an efficient man...

We give a simple argument showing that the number of edges in the intersection graph G of a family of n sets in the plane with a linear unioncomplexity is O(ω(G)n). In particular, we prove χ(G) 6 col(G) < 19ω(G) for intersection graph G of a family of pseudo-discs, which improves a previous

Let R be a ring with unity and I(R) be the set of all non-trivial left ideals of R. The intersection graph of ideals of R, denoted by G(R), is a graph with the vertex set I(R) and two distinct vertices I and J are adjacent if and only if I ∩ J 6= 0. In this paper, we study some connections between the graph-theoretic properties of this graph and some algebraic properties of rings. We characteri...