Intersection number (graph theory)

Abstract: result of the social structure of American society.’’ This is true of the particular form or version of organized crime which he describes— it did not emanate from a transplanted Sicilian Mafia. That explanation, however, does not account for the many faces of organized crime in Australia, China, Russia, Japan, and many other places. Nor does it account for the growing phenomenon of transnational organized crime. Apart from any points of disagreement, this is a serious work of scholarship on a subject that has too often gotten short shrift in that respect. Organized Crime in Chicago is a good addition to the organized crime literature.

Abstract: In this paper we address ourselves to identifying facets of the set packing polyhedron, i.e., of the convex hull of integer solutions to the set covering problem with equality constraints and/or constraints of the form "ź". This is done by using the equivalent node-packing problem derived from the intersection graph associated with the problem under consideration. First, we show that the cliques of the intersection graph provide a first set of facets for the polyhedron in question. Second, it is shown that the cycles without chords of odd length of the intersection graph give rise to a further set of facets. A rather strong geometric property of this set of facets is exhibited.

Abstract: We are concerned here with one of the oldest problems in combinatorial extremal theory. It is readily described after we have made a few conventions. ‫ގ‬ denotes the set of

Abstract: Traditional random graph models of networks generate networks that are locally treelike, meaning that all local neighborhoods take the form of trees. In this respect such models are highly unrealistic, most real networks having strongly nontreelike neighborhoods that contain short loops, cliques, or other biconnected subgraphs. In this paper we propose and analyze a class of random graph models that incorporates general subgraphs, allowing for nontreelike neighborhoods while still remaining solvable for many fundamental network properties. Among other things we give solutions for the size of the giant component, the position of the phase transition at which the giant component appears, and percolation properties for both site and bond percolation on networks generated by the model.