Constructing and classifying neighborhood anti-Sperner graphs

Orphan complexes of neighborhood anti-Sperner graphs

We introduce and study a class of simplicial complexes, the orphan complexes, associated to simple graphs whose family of (open or closed) vertex-neighborhoods are anti-Sperner. Under suitable restrictions, we show that orphan complexes of such graphs are always shellable and provide a characterization of graphs in terms of induced forbidden subgraphs contained in this restricted subfamily.

Enumerating Labelled Graphs with Certain Neighborhood Properties

Properties of (connected) graphs whose closed or open neighborhood families are Sperner, anti-Sperner, distinct or none of the proceeding have been extensively examined. In this paper we examine 24 properties of the neighborhood family of a graph. We give asymptotic formulas for the number of (connected) labelled graphs for 12 of these properties. For the other 12 properties, we give bounds for...

Closed-neighbourhood anti-Sperner graphs

Sperner capacities

We determine the asymptotics of the largest family of qualitatively 2{independent k{ partitions of an n{set, for every k > 2. We generalize a Sperner-type theorem for 2{partite sets of KK orner and Simonyi to the k{partite case. Both results have the feature that the corresponding trivial information-theoretic upper bound is tight. The results follow from a more general Sperner capacity theorem...

Classifying pentavalnet symmetric graphs of order $24p$

A graph is said to be symmetric if its automorphism group is transitive on its arcs. A complete classification is given of pentavalent symmetric graphs of order 24p for each prime p. It is shown that a connected pentavalent symmetric graph of order 24p exists if and only if p=2, 3, 5, 11 or 17, and up to isomorphism, there are only eleven such graphs.