Coloring of integer distance graphs

Kemnitz, Arnfried; Kolberg, Halka

doi:10.1016/s0012-365x(98)00099-5

Cited by 36 publications

(26 citation statements)

References 7 publications

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“…The case that x and y are of distinct parity is easy. It was proved by Kemnitz and Kolberg [17] that for this case, ðGðZ; MÞÞ ¼ 4. Hence, ðGðZ; MÞÞ ¼ !ðGðZ; MÞÞ ¼ 4, so we have Theorem 4.1.…”

Section: Almost Difference Closed Sets Of Type A3mentioning

confidence: 88%

“…Note that by using Theorem 4.1, Lemma 4.1, and (1.2), we can determine the chromatic number for this type of sets M, which was obtained in Refs. [17] and [18] by different approaches. As remarked earlier, ðMÞ !…”

Section: Consequences and Related Problemsmentioning

confidence: 99%

“…The distance graph generated by M, denoted by GðZ; MÞ, has the set Z of all integers as the vertex set, and two vertices x and y are adjacent whenever jx À yj 2 M. Initiated by Eggleton, Erdo }s, and Skilton [10], the study of distance graphs has attracted considerable attention. Most of the efforts in the past are devoted to the chromatic number of such graphs [4,8,[10][11][12][13][17][18][19][20]23,24].…”

Section: Introductionmentioning

confidence: 99%

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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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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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Section: Almost Difference Closed Sets Of Type A3mentioning

confidence: 88%

Section: Consequences and Related Problemsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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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 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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“…For different types of distance sets D, the problem of determining χ (Z , D) has been studied extensively, see Refs [4,5,7,9,10,12,[16][17][18][19]. For instance, the case that D contains at most three integers were studied by Eggleton, Erdős and Skilton [9], Chen, Chang and Huang [5], Voigt [16], Deuber and Zhu [7], Zhu [18], and at last completely determined by Zhu [19].…”

Section: Introductionmentioning

confidence: 99%