Orphan complexes of neighborhood anti-Sperner graphs

Constructing and classifying neighborhood anti-Sperner graphs

For a simple graph G let NG(u) be the (open) neighborhood of vertex u ∈ V (G). Then G is neighborhood anti-Sperner (NAS) if for every u there is a v ∈ V (G)\{u} with NG(u) ⊆ NG(v). And a graph H is neighborhood distinct (ND) if every neighborhood is distinct, i.e., if NH(u) 6= NH(v) when u 6= v, for all u and v ∈ V (H). In Porter and Yucas [3] a characterization of regular NAS graphs was given:...

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...

Convex hulls of more-part Sperner families

The convex hulls of more-part Sperner families is defined and studied. Corollaries of the results are some well-known theorems on 2 or 3-part Sperner families. Some methods are presented giving new theorems.