Uniform distances in rational unit-distance graphs

On -unit distance graphs

We consider a variation on the problem of determining the chromatic number of the Euclidean plane and define the -unit distance graph to be the graph whose vertices are the points of E, in which two points are adjacent whenever their distance is within of 1. For certain values of we are able to show that the chromatic number is exactly seven. For some smaller values we show the chromatic number...

Independent Complementary Distance Pattern Uniform Graphs

A graph G = (V,E) is called to be Smarandachely uniform k-graph for an integer k ≥ 1 if there exists M1,M2, · · · ,Mk ⊂ V (G) such that fMi(u) = {d(u, v) : v ∈ Mi} for ∀u ∈ V (G)−Mi is independent of the choice of u ∈ V (G)−Mi and integer i, 1 ≤ i ≤ k. Each such set Mi, 1 ≤ i ≤ k is called a CDPU set [6, 7]. Particularly, for k = 1, a Smarandachely uniform 1-graph is abbreviated to a complement...

Points at Rational Distance from the Vertices of a Unit Polygon

Unit Distance Graphs with Ambiguous Chromatic Number

First László Székely and more recently Saharon Shelah and Alexander Soifer have presented examples of infinite graphs whose chromatic numbers depend on the axioms chosen for set theory. The existence of such graphs may be relevant to the Chromatic Number of the Plane problem. In this paper we construct a new class of graphs with ambiguous chromatic number. They are unit distance graphs with ver...

Two notions of unit distance graphs

A faithful (unit) distance graph in Rd is a graph whose set of vertices is a finite subset of the d-dimensional Euclidean space, where two vertices are adjacent if and only if the Euclidean distance between them is exactly 1. A (unit) distance graph in Rd is any subgraph of such a graph. In the first part of the paper we focus on the differences between these two classes of graphs. In particula...