Circular Chromatic Numbers and Fractional Chromatic Numbers of Distance Graphs

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 ji À jj 2 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 jMj, 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]).

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“…The following is a corollary of Theorem 2.1, where the a ¼ 1 case was first proved in Ref. [3] and the a ! 2 case was first proved in Ref.…”

Section: Consequences and Related Problemsmentioning

confidence: 86%

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

Liu

Zhu

2004

Journal of Graph Theory

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“…It turns out that many of the star extremal circulant graphs currently known in the literature [1,11,15,17,18] are specific examples of circulants in our infinite family. Thus, our construction is a broad generalization of previously published results.…”

Section: G Is Star Extremalmentioning

confidence: 99%

“…A particularly interesting problem is determining the value of (G(Z, S)) for a given set S. Much work has been done on this problem [1,2,7,11,14,18,19,23].…”

Section: An Application To Integer Distance Graphsmentioning

confidence: 99%

Proof of a conjecture on fractional Ramsey numbers

Brown

Hoshino

2009

Journal of Graph Theory

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Jacobson, Levin, and Scheinerman introduced the fractional Ramsey function r f (a 1 , a 2 , . . . , a k ) as an extension of the classical definition for Ramsey numbers. They determined an exact formula for the fractional Ramsey function for the case k = 2. In this article, we answer an open problem by determining an explicit formula for the general case k>2 by constructing an infinite family of circulant graphs for which the independence numbers can be computed explicitly. This construction gives us two further results: a new (infinite) family of star extremal graphs which are a superset of many of the families currently known in the literature, and a broad generalization of known results on the chromatic number of integer distance graphs. ᭧

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“…A fractional coloring of a graph G is a mapping h from I(G), the set of all independent sets of G, to the interval [0,1] …”

Section: Chromatic Number Of G(z D) Is Denoted By χ(Z D)mentioning

confidence: 99%

“…Now we assume that [1,11,19], and the value of χ c (Z, D m,k,s ) has been completely determined in [19]. (The circular chromatic number χ c (G) of a graph G is a refinement of χ(G), and χ(G) = χ c (G) for any graph G. For a survey of research concerning circular chromatic number of graphs, see [20].…”

Section: Proof It Suffices To Show Thatmentioning

confidence: 99%

Distance graphs with missing multiples in the distance sets

Liu

Zhu

1999

J. Graph Theory

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Given positive integers m, k and s with m > ks, let D m,k,s represent the set {1, 2, • • • , m} − {k, 2k, • • • , sk}. The distance graph G(Z, D m,k,s ) has as vertex set all integers Z and edges connecting i and j whenever |i − j| ∈ D m,k,s . The chromatic number and the fractional chromatic number of G(Z, D m,k,s ) are denoted by χ(Z, D m,k,s ) and χ f (Z, D m,k,s ), respectively. For s = 1, χ(Z, D m,k,1 ) was studied by Eggleton, Erdős and Skilton [6], Kemnitz and Kolberg [12], and Liu [13], and was solved lately by Chang, Liu and Zhu [2] who also determined χ f (Z, D m,k,1 ) for any m and k. This article extends the study of χ(Z, D m,k,s ) and χ f (Z, D m,k,s ) to general values of s. We proveThe latter result provides a good lower bound for χ(Z, D m,k,s ). A general upper bound for χ(Z, D m,k,s ) is found. We prove the upper bound can be improved to (m + sk + 1)/(s + 1) + 1 for some values of m, k and s. In particular, when s + 1 is prime, χ(Z, D m,k,s ) is either (m + sk + 1)/(s + 1) or (m + sk + 1)/(s + 1) + 1. By using a special coloring method called the pre-coloring method, many distance graphs G(Z, D m,k,s ) are classified into * Supported in part by the National Science Foundation under grant DMS 9805945. † Supported in part by the National Science Council under grant NSC87-2115-M110-004.these two possible values of χ(Z, D m,k,s ). Moreover, complete solutions of χ(Z, D m,k,s ) for several families are determined including the case s = 1 (solved in [2]), the case s = 2, the case (k, s + 1) = 1, and the case that k is a power of a prime.

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Circular Chromatic Numbers and Fractional Chromatic Numbers of Distance Graphs

Cited by 35 publications

References 14 publications

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

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

Proof of a conjecture on fractional Ramsey numbers

Distance graphs with missing multiples in the distance sets

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