Peter Johnson

Integer and fractional security in graphs

Let G=(V,E) be a graph. A subset S of V is said to be secure if it can defend itself from an attack by vertices in N[S]-S. In the usual definition, each vertex in N[S]-S can attack exactly one vertex in S, and each vertex in S can defend itself or one of its neighbors in S. A defense of S is successful if each vertex has as many defenders as attackers. We look at allowing an attacking vertex to divide its one unit of attack among multiple targets, or allowing a defending vertex to divide its one unit of defense among multiple allies. Three new definitions of security are given. It turns out that two of the new definitions are the same as the original.

Two-colorings of a dense subgroup of Qn that forbid many distances

Abstract For any positive integer n, there is a two-coloring of the subgroup of Q n consisting of rational points with “odd denominator” which forbids all (usual Euclidean) distances √(p/q), p, q odd positive integers. From this it follows that there are a two-coloring of Q 3, and a four-coloring of Q 4, which forbid those distances. These results improve the known results that the chromatic numbers of Q 3 and Q 4 are 2 and 4, respectively.

SHIELDS-HARARY NUMBERS OF GRAPHS WITH RESPECT TO CONTINUOUS CONCAVE COST FUNCTIONS

The Shields-Harary numbers are a class of graph parameters that measure a certain kind of robustness of a graph, thought of as a network of fortified reservoirs, with reference to a given cost function. We prove a result about the Shields-Harary numbers with respect to concave continuous cost functions which will simplify the calculation of these numbers for certain classes of graphs, including graphs formed by two intersecting cliques, and complete multipartite graphs.

Reverse class critical multigraphs

Abstract We characterize simple graphs G which are Class 1 and have the property that G \υ is Class 2 (∀ υ ∈ V ( G )). We also discuss multigraphs with this property.

Simple product colorings

If A is a set colored with m colors, and B is colored with n colors, the coloring of A x B obtained by coloring (a, b) with the pair (color of a, color of b) will be called an m x n simple product coloring (SPC) of A x B. SPCs of Cartesian products of three or more sets are defined analogously. It is shown that there are 2 x 2, and 2 x 2 x 2 SPCs of Q^2 and Q^3 which forbid the distance one; that there is no 2^k SPC of Q^k forbidding the distance one, for k > 3; and that there is no 2 x 2 SPC of Q x Q(@/15), and thus none of R^2, forbidding the distance 1.

Euclidean Distance Graphs on the Rational Points

Throughout, \(\mathbb{Z}\), \(\mathbb{Q}\), and \(\mathbb{R}\) denote the usual rings of integers, rational numbers, and real numbers, respectively. If X is a set and n is a positive integer, X n denotes, as usual, the set of n-tuples with entries from X.

Szlam’s Lemma: Mutant Offspring of a Euclidean Ramsey Problem from 1973, with Numerous Applications

If you treasure semantic precision, you might name the volcano that erupted (figuratively speaking!) in 1973: Coloring Problems in Geometrically Defined Hypergraphs.