Two Characterizations of Hypercubes

Abstract

Two characterizations of hypercubes are given: 1) A graph is a hypercube if and only if it is antipodal and bipartite $(0,2)$-graph. 2) A graph is an $n$-hypercube if and only if there are $n$ pairs of prime convexes, the graph is a prime convex intersection graph, and each intersection of $n$ prime convexes (no one of which is from the same pair) is a vertex.

4-cycle properties for characterizing\newline rectagraphs and hypercubes

“…for any two vertices u and v in V (G). Note that a hypercube is characterized as a (0, 2)-graph, see [5], [6], interval monotone [4], interval regular [8] and an antipodal graph [10].…”

“…(0, 2)-graphs were studied in various contexts; existence and construction of (0, 2)graphs were intensively studied by several researchers, we can cite for instance [2], [3]. Another important aspect is characterizing hypercubes as (0, 2)-graphs; due to their remarkable properties and multiple applications, many authors have investigated this topic in order to give a new point of view which can be used for recognizing and constructing hypercubes, see [1], [5], [6], [7], [10].…”