Number Of Distance Graphs Research Articles

On a Frankl-Wilson Theorem

We derive an analog of the Frankl-Wilson theorem on independence numbers of some distance graphs. The obtained results are applied to the problem of the chromatic number of a space ℝn with a forbidden equilateral triangle and to the problem of chromatic numbers of distance graphs with large girth.

On the Independence Numbers of Distance Graphs with Vertices in {–1, 0, 1}n

On the Independence Numbers of Distance Graphs with Vertices in {–1, 0, 1}n

On the Independence Numbers of Some Distance Graphs with Vertices in {–1, 0, 1}n*

New estimates for the independence numbers of distance graphs with vertices in $${{{\text{\{}}{\kern 1pt} - {\kern 1pt} 1,0,1{\text{\} }}}^{n}}$$ are obtained.

New upper bound for the chromatic number of a random subgraph of a distance graph

This paper is related to the classical Hadwiger-Nelson problem dealing with the chromatic numbers of distance graphs in ℝn. We consider the class consisting of the graphs G(n, 2s + 1, s) = (V (n, 2s + 1), E(n, 2s + 1, s)) defined as follows: $$V(n,2s + 1) = \{ x = (x_1 ,x_2 ,...,x_n ):x_i \in \{ 0,1\} ,x_1 + x_2 + \cdots + x_n = 2s + 1\} ,E(n,2s + 1,s) = \{ \{ x,y\} :(x,y) = s\} ,$$ where (x, y) stands for the inner product. We study the random graph G(G(n, 2s + 1, s), p) each of whose edges is taken from the set E(n, 2s+1, s) with probability p independently of the other edges. We prove a new bound for the chromatic number of such a graph.

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 particular, we show that for any fixed d the number of faithful distance graphs in Rd on n labelled vertices is 2(1+o(1))dnlog2n, and give a short proof of the known fact that the number of distance graphs in Rd on n labelled vertices is 2(1−1/⌊d/2⌋+o(1))n2/2. We also study the behavior of several Ramsey-type quantities involving these graphs.In the second part of the paper we discuss the problem of determining the minimum possible number of edges of a graph which is not isomorphic to a faithful distance graph in Rd.

New lower bounds for the independence numbers of distance graphs with vertices in {−1, 0, 1} n

New lower bounds for the independence numbers of distance graphs with vertices in {−1, 0, 1} n

On independence numbers of distance graphs with vertices in {-1,0,1} n : estimates, conjectures, and applications to the Nelson–Erdős–Hadwiger problem and the Borsuk problem

The paper states and studies a problem that is closely related to the problems mentioned in the title.

On the independence number of distance graphs with vertices in {−1, 0, 1} n

New estimates for the independence numbers of distance graphs with vertices in B {-1, 0, 1} n are obtained.

Regular chromatic number and the lonely runner problem

Abstract Suppose k + 1 runners having nonzero constant speeds run laps on a unit-length circular track. The lonely runner conjecture states that there is a time at which one runner is at distance at least 1 / ( k + 1 ) from all the others. The conjecture has been proved up to six runners ( k ⩽ 5 ) . A formulation of the problem is related to the regular chromatic number of distance graphs. We use a new tool developed in this context to solve the first open case of the conjecture with seven runners.