Isoperimetric Numbers of Randomly Perturbed Intersection Graphs

Regular Honest Graphs, Isoperimetric Numbers, and Bisection of Weighted Graphs

The edge-integrity of a graph G is I 0(G) := minfjSj + m(G S) : S Eg; where m(H) denotes the maximum order of a component of H: A graph G is called honest if its edge-integrity is the maximum possible, that is, equals the order of the graph. The only honest 2-regular graphs are the 3-, 4-, and 5-cycles. Lipman [13] proved that there are exactly twenty honest cubic graphs. In this paper we explo...

Lower Bounds for the Isoperimetric Numbers of Random Regular Graphs

The vertex isoperimetric number of a graph G = (V ,E) is the minimum of the ratio |∂V U |/|U | where U ranges over all nonempty subsets of V with |U |/|V | ≤ u and ∂V U is the set of all vertices adjacent to U but not in U . The analogously defined edge isoperimetric number—with ∂V U replaced by ∂EU , the set of all edges with exactly one endpoint in U —has been studied extensively. Here we stu...

Isoperimetric Numbers of Cayley Graphs Arising from Generalized Dihedral Groups

Let n, x be positive integers satisfying 1 < x < n. Let Hn,x be a group admitting a presentation of the form 〈a, b | a = b = (ba) = 1〉. When x = 2 the group Hn,x is the familiar dihedral group, D2n. Groups of the form Hn,x will be referred to as generalized dihedral groups. It is possible to associate a cubic Cayley graph to each such group, and we consider the problem of finding the isoperimet...

Bounded-Degree Spanning Trees in Randomly Perturbed Graphs

We show that for any xed dense graph G and bounded-degree tree T on the same number of vertices, a modest random perturbation of G will typically contain a copy of T . This combines the viewpoints of the well-studied problems of embedding trees into xed dense graphs and into random graphs, and extends a sizeable body of existing research on randomly perturbed graphs. Speci cally, we show that t...

Expansion and Lack Thereof in Randomly Perturbed Graphs