An intriguing theme in graph theory is that of the intersection graph of a family of subsets of a set: the vertices of the graph are represented by the subsets of the family and adjacency is defined by a non-empty intersection of the corresponding subsets. Prime examples are interval graphs and chordal graphs. An interval graph is the intersection graph of a family of closed intervals on the re...

We study bipartite geometric intersection graphs from the perspective of order dimension. We show that partial orders of height two whose comparability graph is a grid intersection graph have order dimension at most four. Starting from this observation we look at various classes of graphs between grid intersection graph and bipartite permutations graphs and the containment relation on these cla...

The sphericity sph(G) of a graph G is the minimum dimension d for which G is the intersection graph of a family of congruent spheres in Rd . The edge clique cover number (G) is the minimum cardinality of a set of cliques (complete subgraphs) that covers all edges of G. We prove that if G has at least one edge, then sph(G) (G). Our upper bound remains valid for intersection graphs defined by bal...

In this paper we study restrictions imposed on individual preferences, as opposed to preference profiles, and we introduce individual preference restrictions that characterize the existence and uniqueness of stable hedonic coalition structures on rich preference domains. We prove that if such a preference domain satisfies an individual preference restriction then it guarantees the existence of ...

We prove a sufficient condition for a graph G to have a matching that interconnects all the components of a disconnected spanning subgraph ofG. The condition is derived from a recent extension of the Matroid intersection theorem due to Aharoni and Berger. We apply the result to the problem of the existence of a (spanning) 2-walk in sufficiently tough graphs.

Assume that G = G(V, E) is an undirected graph with vertex set V and edge set E. A clique of G is a complete subgraph. An edge clique-covering is a family of cliques of G which cover all edges of G. The edge clique-cover number, Be( G), is the minimum number of cliques in an edge clique-cover of G. For results and applications of the edge clique-cover number see [1-4]. Observe that Be(G) does n...

Given a combinatorial designD with block set B, its traditional block-intersection graph GD is the graph having vertex set B such that two vertices b1 and b2 are adjacent if and only if b1 and b2 have non-empty intersection. In this paper we consider the S-block-intersection graph, in which two vertices b1 and b2 are adjacent if and only if |b1 ∩ b2| ∈ S. As our main result we prove that {1, 2,...

Abstract: The intersection dimension of a graph G with respect to a class A of graphs is the minimum k such that G is the intersection of some k graphs on the vertex set V (G) belonging to A. In this paper we follow [ Kratochv́ıl J., Tuza Z.: Intersection dimensions of graph classes, Graphs and Combinatorics 10 (1994), 159– 168 ] and show that for some pairs of graph classes A, B the intersectio...

In this article we consider the intersection graph G(R) of nontrivial proper ideals of a finite commutative principal ideal ring R with unity 1. Two distinct ideals are adjacent if they have non-trivial intersection. We characterize when the intersection graph is complete, bipartite, planar, Eulerian or Hamiltonian. We also find a formula to calculate the number of ideals in each ring and the d...

A graph is ptolemaic if and only if it is both chordal and distancehereditary. Thus, a ptolemaic graph G has two kinds of intersection graph representations: one from being chordal, and the other from being distance-hereditary. The first of these, called a clique tree representation, is easily generated from the clique graph of G (the intersection graph of the maximal complete subgraphs of G). ...