Circular chromatic numbers of some distance graphs

AbstractGiven a set D of positive integers, the distance graph G(Z,D) has vertices all integers Z, and two vertices j and j′ in Z are adjacent if and only if |j-j′|∈D. This paper determines the circular chromatic numbers of some distance graphs

Circular chromatic numbers of a class of distance graphs

AbstractSuppose m,k,s are positive integers with m>sk. Let Dm,k,s denote the set {1,2,…,m}⧹{k,2k,…,sk}. The distance graph G(Z,Dm,k,s) has as vertex set all integers Z and edges connecting i and j whenever |i−j|∈Dm,k,s. This paper determines the circular chromatic number of all the distance graphs G(Z,Dm,k,s)

Connectivity and diameter in distance graphs

For $n\in \mathbb{N}$ and $D\subseteq \mathbb{N}$, the distance graph $P_n^D$ has vertex set $\{ 0,1,\ldots,n-1\}$ and edge set $\{ ij\mid 0\leq i,j\leq n-1, |j-i|\in D\}$. The class of distance graphs generalizes the important and very well-studied class of circulant graphs which have been proposed for numerous network applications. In view of fault tolerance and delay issues in these applications, the connectivity and diameter of circulant graphs have been studied in great detail. Our main contributions are hardness results concerning computational problems related to the connectivity and diameter of distance graphs and a number-theoretic characterization of the connected distance graphs $P_n^D$ for $|D|=2$

On the independence ratio of distance graphs

A distance graph is an undirected graph on the integers where two integers are adjacent if their difference is in a prescribed distance set. The independence ratio of a distance graph $G$ is the maximum density of an independent set in $G$. Lih, Liu, and Zhu [Star extremal circulant graphs, SIAM J. Discrete Math. 12 (1999) 491--499] showed that the independence ratio is equal to the inverse of the fractional chromatic number, thus relating the concept to the well studied question of finding the chromatic number of distance graphs. We prove that the independence ratio of a distance graph is achieved by a periodic set, and we present a framework for discharging arguments to demonstrate upper bounds on the independence ratio. With these tools, we determine the exact independence ratio for several infinite families of distance sets of size three, determine asymptotic values for others, and present several conjectures.Comment: 39 pages, 12 figures, 6 table

Powers of cycles, powers of paths, and distance graphs

In 1988, Golumbic and Hammer characterized powers of cycles, relating them to circular-arc graphs. We extend their results and propose several further structural characterizations for both powers of cycles and powers of paths. The characterizations lead to linear-time recognition algorithms of these classes of graphs. Furthermore, as a generalization of powers of cycles, powers of paths, and even of the well-known circulant graphs, we consider distance graphs. While colourings of these graphs have been intensively studied, the recognition problem has been so far neglected. We propose polynomial-time recognition algorithms for these graphs under additional restrictions

On the density of integral sets with missing differences from sets related to arithmetic progressions

For a given set M of positive integers, a problem of Motzkin asks for determining the maximal density μ(M) among sets of nonnegative integers in which no two elements differ by an element of M. The problem is completely settled when |M| 2, and some partial results are known for several families of M for |M| 3, including the case where the elements of M are in arithmetic progression. We consider some cases when M either contains an arithmetic progression or is contained in an arithmetic progression

Coloring of Integer Distance Graphs

. An integer distance graph is a graph G(D) with the set of integers as vertex set and with an edge joining two vertices u and v if and only if ju \Gamma vj 2 D where D is a subset of the positive integers. We determine the chromatic number (D) of G(D) for some finite distance sets D such as sets of consecutive integers and special sets of cardinality 4. 1. Introduction For any D ` IN with IN the set of all positive integers let G(D) denote the graph with the set ZZ of integers as vertex set and with an edge joining two vertices u and v if and only if ju \Gamma vj 2 D. Such a graph G(D) is called integer distance graph or simply distance graph (of the distance set D). A coloring f : V(G) ! ff 1 ; : : : ; f k g of G is an assignment of colors to the vertices of G such that f(u) 6= f(v) for all adjacent vertices u and v. The minimum number of colors necessary to color G is the chromatic number (G). In this paper we study the chromatic number (G(D)) = (D) = (d 1 ; d 2 ; : : :) of ..