Circular Chromatic Numbers of Distance Graphs with Distance Sets Missing Multiples

Huang, Lingling; Chang, Gerard J.

doi:10.1006/eujc.1999.0284

Cited by 13 publications

(13 citation statements)

References 13 publications

(11 reference statements)

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“…The circular chromatic numbers of distance graphs have been studied in [3,4,14,21]. If D contains two integers, then the circular chromatic number of the distance graph GðR; DÞ was determined in [3].…”

Section: Introductionmentioning

confidence: 99%

Circular chromatic number of distance graphs with distance sets of cardinality 3

Zhu

2002

Journal of Graph Theory

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Suppose D is a subset of R þ . The distance graph G(R, D) is the graph with vertex set R in which two vertices x,y are adjacent if jx À y j 2 D. This study investigates the circular chromatic number and the fractional chromatic number of distance graphs G(R, D) with jDj ¼ 3. As a consequence, we determine the chromatic numbers of all such distance graphs. This settles a conjecture proposed independently by Chen et al.

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Section: Introductionmentioning

confidence: 99%

Circular chromatic number of distance graphs with distance sets of cardinality 3

Zhu

2002

Journal of Graph Theory

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“…It follows from the definition that ψ(i) = j if and only if ψ(i + [3,7,10,18,19,24,25,33]. Erdős, Eggleton and Skelton initiated the study of this family of distance graphs for the case that s = 1 = [10].…”

Section: Proof It Is Not Difficult To See That Whenmentioning

confidence: 99%

Asymptotic Clique Covering Ratios of Distance Graphs

Liu

Zhu

2002

European Journal of Combinatorics

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Given a finite set D of positive integers, the distance graph G(Z , D) has Z as the vertex set and {i j : |i − j| ∈ D} as the edge set. Given D, the asymptotic clique covering ratio is defined as (Z , D)), where ω(G) is the clique number of G.) This problem turns out to be related to T -colorings and to fractional chromatic number and circular chromatic number of distance graphs. Through such connections, we shall show that the equality S(D) = ω(G(Z , D)) holds for many classes of distance graphs. Moreover, we raise questions regarding other such connections.

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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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Circular Chromatic Numbers of Distance Graphs with Distance Sets Missing Multiples

Cited by 13 publications

References 13 publications

Circular chromatic number of distance graphs with distance sets of cardinality 3

Circular chromatic number of distance graphs with distance sets of cardinality 3

Asymptotic Clique Covering Ratios of Distance Graphs

Proof of a conjecture on fractional Ramsey numbers

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