Chromatic Numbers Of Distance Graphs Research Articles

Chromatic Numbers of Distance Graphs without Short Odd Cycles in Rational Spaces

Chromatic Numbers of Distance Graphs without Short Odd Cycles in Rational Spaces

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 Frankl–Rödl theorem

In this paper we considerably strengthen the currently known explicit exponential lower bound for the chromatic number of a Euclidean space with a forbidden regular simplex. We also strengthen the exponential lower bound for the chromatic numbers of distance graphs with large girth.

Improved Frankl–Rödl Theorem and Some of Its Geometric Consequences

We substantially improve a presently known explicit exponentially growing lower bound on the chromatic number of a Euclidean space with forbidden equilateral triangle. Furthermore, we improve an exponentially growing lower bound on the chromatic number of distance graphs with large girth. These refinements are obtained by improving known upper bounds on the product of cardinalities of two families of homogeneous subsets with one forbidden cross-intersection.

Chromatic Numbers of Distance Graphs with Several Forbidden Distances and without Cliques of a Given Size

We consider distance graphs with k forbidden distances in an n-dimensional space with the p-norm that do not contain cliques of a fixed size. Using a probabilistic construction, we present graphs of this kind with chromatic number at least (Bk) Cn , where B and C are constants.

Lower bounds for the chromatic numbers of distance graphs with large girth

We obtain some specific exponential lower bounds for the chromatic numbers of distance graphs with large girth.

Lower Bounds for the Chromatic Numbers of Distance Graphs with Large Girth

Получены некоторые конкретные нижние экспоненциальные оценки хроматических чисел дистанционных графов с большим обхватом. Библиография: 30 названий.

Independence numbers and chromatic numbers of the random subgraphs of some distance graphs

This work is concerned with the Nelson-Hadwiger classical problem of finding the chromatic numbers of distance graphs in . Most consideration is given to the class of graphs defined as follows: where is the Euclidean inner product. In particular, the chromatic number of was recently found by Balogh, Kostochka and Raigorodskii. The objects of the study are the random subgraphs whose edges are independently taken from the set , each with probability . The independence number and the chromatic number of such graphs are estimated both from below and from above. In the case when is a prime power and , the order of growth of and is established. Bibliography: 51 titles.

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.

New upper bounds for the independence numbers of graphs with vertices in {−1, 0, 1} n and their applications to problems of the chromatic numbers of distance graphs

Upper bounds for the independence numbers in the graphs with vertices at {−1, 0, 1} n are improved. Their applications to problems of the chromatic numbers of distance graphs are studied.

VERTEX COLORING OF CERTAIN DISTANCE GRAPHS

In this paper first, we give a brief introduction about integer distance graphs. An integer distance graph is a graph G(Z,D) with the set of integers as vertex set and an edge joining two vertices u and v if and only if |u − v| ∈ D where D is a subset of the positive integers. If D is a subset of P then we call G(Z,D) a prime distance graph. Second, we obtain a partial solution to a general open problem of characterizing a class of prime distance graphs. Third, we compute the vertex arboricity of certain prime distance graphs. Fourth, we give a brief review regarding circulant graphs and highlight its importance in the computation of chromatic number of distance graphs with appropriate references. Fifth, we introduce the notion of pseudochromatic coloring and obtain certain results concerning circulant graphs and distance graphs.

Chromatic number of distance graphs generated by the sets {2,3,x,y}

Let D be a set of positive integers. The distance graph generated by D has all integers ℤ as the vertex set; two vertices are adjacent whenever their absolute difference falls in D. We completely determine the chromatic number for the distance graphs generated by the sets D={2,3,x,y} for all values x and y. The methods we use include the density of sequences with missing differences and the parameter involved in the so called “lonely runner conjecture”. Previous results on this problem include: For x and y being prime numbers, this problem was completely solved by Voigt and Walther (Discrete Appl. Math. 51:197–209, 1994); and other results for special integers of x and y were obtained by Kemnitz and Kolberg (Discrete Math. 191:113–123, 1998) and by Voigt and Walther (Discrete Math. 97:395–397, 1991).

Packing chromatic number of distance graphs

The packing chromatic number χρ(G) of a graph G is the smallest integer k such that vertices of G can be partitioned into disjoint classes X1,…,Xk where vertices in Xi have pairwise distance greater than i. We study the packing chromatic number of infinite distance graphs G(Z,D), i.e., graphs with the set Z of integers as vertex set and in which two distinct vertices i,j∈Z are adjacent if and only if |i−j|∈D.In this paper we focus on distance graphs with D={1,t}. We improve some results of Togni who initiated the study. It is shown that χρ(G(Z,D))≤35 for sufficiently large odd t and χρ(G(Z,D))≤56 for sufficiently large even t. We also give a lower bound 12 for t≥9 and tighten several gaps for χρ(G(Z,D)) with small t.

Lower Bounds for Measurable Chromatic Numbers

The Lovász theta function provides a lower bound for the chromatic number of finite graphs based on the solution of a semidefinite program. In this paper we generalize it so that it gives a lower bound for the measurable chromatic number of distance graphs on compact metric spaces. In particular we consider distance graphs on the unit sphere. There we transform the original infinite semidefinite program into an infinite linear program which then turns out to be an extremal question about Jacobi polynomials which we solve explicitly in the limit. As an application we derive new lower bounds for the measurable chromatic number of the Euclidean space in dimensions 10, . . . , 24 and we give a new proof that it grows exponentially with the dimension.

FROM RAINBOW TO THE LONELY RUNNER: A SURVEY ON COLORING PARAMETERS OF DISTANCE GRAPHS

Motivated by the plane coloring problem, Eggleton, Erd\H{o}s and Skelton initiated the study of distance graphs. Let $D$ be a set of positive integers. The distance graph generated by $D$, denoted by $G(\mathbb{Z}, D)$, has all integers $\mathbb{Z}$ as the vertex set, and two vertices $x$ and $y$ are adjacent whenever $|x-y| \in D$. The chromatic number, circular chromatic number and fractional chromatic number of distance graphs have been studied extensively in the past two decades; these coloring parameters are also closely related to some problems studied in number theory and geometry. We survey some research advances and open problems on coloring parameters of distance graphs.

The Lonely Runner with Seven Runners

Suppose $k+1$ runners having nonzero constant pairwise distinct speeds run laps on a unit-length circular track starting at the same time and place. A runner is said to be lonely if she is at distance at least $1/(k+1)$ along the track to every other runner. The lonely runner conjecture states that every runner gets lonely. The conjecture has been proved up to six runners ($k\le 5$). 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.

Fractional chromatic number of distance graphs generated by two-interval sets

Let D be a set of positive integers. The distance graph generated by D , denoted by G ( Z , D ) , has the set Z of all integers as the vertex set, and two vertices x and y are adjacent whenever | x − y | ∈ D . For integers 1 < a ≤ b < m − 1 , define D a , b , m = { 1 , 2 , … , a − 1 } ∪ { b + 1 , b + 2 , … , m − 1 } . For the special case a = b , the chromatic number for the family of distance graphs G ( Z , D a , a , m ) was first studied by R.B. Eggleton, P. Erdős and D.K. Skilton [Colouring the real line, J. Combin. Theory (B) 39 (1985) 86–100] and was completely solved by G. Chang, D. Liu and X. Zhu [Distance graphs and T -coloring, J. Combin. Theory (B) 75 (1999) 159–169]. For the general case a ≤ b , the fractional chromatic number for G ( Z , D a , b , m ) was studied by P. Lam and W. Lin [Coloring distance graphs with intervals as distance sets, European J. Combin. 26 (2005) 25 1216–1229] and by J. Wu and W. Lin [Circular chromatic numbers and fractional chromatic numbers of distance graphs with distance sets missing an interval, Ars Combin. 70 (2004) 161–168], in which partial results for special values of a , b , m were obtained. In this article, we completely settle this problem for all possible values of a , b , m .

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.

Using equivariant obstruction theory in combinatorial geometry

A significant group of problems coming from the realm of combinatorial geometry can be approached fruitfully through the use of Algebraic Topology. From the first such application to Kneser's problem in 1978 by Lovász [L. Lovász, Knester's conjecture, chromatic number of distance graphs on the sphere, Acta. Sci. Math (Szeged) 45 (1983) 317–323] through the solution of the Lovász conjecture [E. Babson, D. Kozlov, Proof of Lovasz conjecture, Annals of Mathematics (2) (2004), submitted for publication; C. Schultz, A short proof of w 1 n ( Hom ( C 2 r + 1 , K n + 2 ) ) = 0 for all n and a graph colouring theorem by Babson and Kozlov, 2005, arXiv: math.AT/0507346v2], many methods from Algebraic Topology have been developed. Specifically, it appears that the understanding of equivariant theories is of the most importance. The solution of many problems depends on the existence of an elegantly constructed equivariant map. A variety of results from algebraic topology were applied in solving these problems. The methods used ranged from well known theorems like Borsuk–Ulam and Dold theorem to the integer/ideal-valued index theories. Recently equivariant obstruction theory has provided answers where the previous methods failed. For example, in papers [R.T. Živaljević, Equipartitions of measures in R 4 , arXiv: math.0412483, Trans. Amer. Math. Soc., submitted for publication] and [P. Blagojević, A. Dimitrijević Blagojević, Topology of partition of measures by fans and the second obstruction, arXiv: math.CO/0402400, 2004] obstruction theory was used to prove the existence of different mass partitions. In this paper we extract the essence of the equivariant obstruction theory in order to obtain an effective general position map scheme for analyzing the problem of existence of equivariant maps. The fact that this scheme is useful is demonstrated in this paper with three applications: (A) a “half-page” proof of the Lovász conjecture due to Babson and Kozlov [E. Babson, D. Kozlov, Proof of Lovasz conjecture, Annals of Mathematics (2) (2004), submitted for publication] (one of two key ingredients is Schultz' map [C. Schultz, A short proof of w 1 n ( Hom ( C 2 r + 1 , K n + 2 ) ) = 0 for all n and a graph colouring theorem by Babson and Kozlov, 2005, arXiv: math.AT/0507346v2]), (B) a generalization of the result of V. Makeev [V.V. Makeev, Equipartitions of continuous mass distributions on the sphere an in the space, Zap. Nauchn. Sem. S.-Petersburg (POMI) 252 (1998) 187–196 (in Russian)] about the sphere S 2 measure partition by 3-planes (Section 3), and (C) the new ( a , b , a ) , class of 3-fan 2-measures partitions (Section 3). These three results, sorted by complexity, share the spirit of analyzing equivariant maps from spheres to complements of arrangements of subspaces.

Fractional chromatic number and circular chromatic number for distance graphs with large clique size

AbstractAn Erratum has been published for this article in Journal of Graph Theory 48: 329–330, 2005.Let M be a set of positive integers. The distance graph generated by M, denoted by G(Z, M), has the set Z of all integers as the vertex set, and edges ij whenever |i−j| ∈ M. We investigate the fractional chromatic number and the circular chromatic number for distance graphs, and discuss their close connections with some number theory problems. In particular, we determine the fractional chromatic number and the circular chromatic number for all distance graphs G(Z, M) with clique size at least |M|, except for one case of such graphs. For the exceptional case, a lower bound for the fractional chromatic number and an upper bound for the circular chromatic number are presented; these bounds are sharp enough to determine the chromatic number for such graphs. Our results confirm a conjecture of Rabinowitz and Proulx 22 on the density of integral sets with missing differences, and generalize some known results on the circular chromatic number of distance graphs and the parameter involved in the Wills' conjecture 26 (also known as the “lonely runner conjecture” 1). © 2004 Wiley Periodicals, Inc. J Graph Theory 47: 129–146, 2004