Sequences of integers with missing differences

doi:10.1016/0097-3165(73)90003-4

Journal Article•DOI•

Sequences of integers with missing differences

David G. Cantor¹, Basil Gordon¹•Institutions (1)

University of California, Los Angeles¹

01 May 1973-Journal of Combinatorial Theory, Series A (Academic Press)-Vol. 14, Iss: 3, pp 281-287

TL;DR: The problem is solved for |M| ⩽ 2, and some partial results are obtained in the general case.

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About: This article is published in Journal of Combinatorial Theory, Series A.The article was published on 1973-05-01 and is currently open access. It has received 40 citations till now. The article focuses on the topics: Quadratic integer & Eisenstein integer.

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Journal Article•DOI•

Distance Graphs andT-Coloring

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Gerard J. Chang¹, Daphne Der-Fen Liu², Xuding Zhu³•Institutions (3)

National Chiao Tung University¹, California State University, Los Angeles², National Sun Yat-sen University³

01 Mar 1999-Journal of Combinatorial Theory, Series B

TL;DR: It is proved that for any finite integral setT that contains 0, the asymptoticT-coloring ratioR(T) is equal to the fractional chromatic number of the distance graphG(Z,D), whereD=T?{0}.

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45 citations

Journal Article•DOI•

Circular Chromatic Numbers and Fractional Chromatic Numbers of Distance Graphs

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Gerard J. Chang¹, Lingling Huang¹, Xuding Zhu²•Institutions (2)

National Chiao Tung University¹, National Sun Yat-sen University²

01 May 1998-The Journal of Combinatorics

TL;DR: These numbers for those distance sets D of size two are determined for some specialDsets of size three for D= {1, 2,···,m,n} with 1 ?

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Abstract: This paper studies circular chromatic numbers and fractional chromatic numbers of distance graphsG(Z,D) for various distance setsD. In particular, we determine these numbers for thoseDsets of size two, for some specialDsets of size three, forD= {1, 2,···,m,n} with 1 ?m < n, forD= {q,q+1,···,p} withq

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38 citations

Journal Article•DOI•

The channel assignment problem for mutually adjacent sites

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Jerrold R. Griggs¹, Daphne Der-Fen Liu²•Institutions (2)

University of South Carolina¹, California State University, Los Angeles²

01 Oct 1994-Journal of Combinatorial Theory, Series A

TL;DR: It is proved that the difference sequence {σn + 1 − σn}n = 1∞ is eventually periodic for any T and the greedy (first-fit) T-coloring of Kn also leads to an eventually periodic sequence.

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35 citations

Journal Article•DOI•

Sets of integers with missing differences

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Nicolas M Haralambis

01 Jul 1977-Journal of Combinatorial Theory, Series A

TL;DR: This paper deals with the problem of finding the maximal density μ ( M ) of sets of integers in which the differences given by a set M do not occur, and some general estimates are given.

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30 citations

Journal Article•DOI•

On infinite-difference sets

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Cameron L. Stewart, Robert Tijdeman

01 Oct 1979-Canadian Journal of Mathematics

TL;DR: In this article, it was shown that the infinite difference set of non-negative integers has positive lower density and does not contain gaps of arbitrary length, which is a consequence of Theorem 6.

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Abstract: Let A. be an infinite, strictly increasing sequence of non-negative 1 integers with d(A.) > 0 for i = I, ... ,h. Let the infinite difference set 1 D. of A. be the set of non-negative integers which occur infinitely often 1 1 as the difference of two terms of A .. This paper gives several results on 1 infinite difference sets, thereby answering some questions posed by Erdos. It follows from Theorems I and 2 that D1n ... nDh has positive lower density and does not contain gaps of arbitrary length. There exists even a sequence A with ~(A) > 0 whose infinite difference set equals D1n ... nDh. Theorem 4 says that the collection of infinite difference sets associated with sequences of positive upper density is a filter on the set of all subsets of the non-negative integers. It follows from Theorem 6 that an infinite difference set need not contain an infinite arithmetical progression. Theorems 7 and 8 are related to a problem of Motzkin. He asked how dense a sequence A can be if its difference set does not contain any elements from a given set K. It is a consequence of Theorem 8 that if k 1,k2 , •.• is a sequence of positive integers such that l· k./k. h <~for some positive integer h, J J J+ then there exists a sequence A with d(A) > 0 such that k. is not contained J in the difference set of A for j = 1,2, .•.• All proofs in the paper are elementary and self-contained. Further most results are quantitative; for example, in the cases above where it is stated that E_(A) > 0 we in fact give explicit lower bounds for E_(A).

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28 citations

Cites background from "Sequences of integers with missing ..."

...Cantor and Gordon [1] and more recently Haralambis [4], have obtained some results in this connection, mainly for finite sets K....
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...J+ J J~ then there exists a sequence A of positive upper density such that k. i V(A) J for j = 1,2, •••• In order to prove this conjecture it suffices, by Theorems I and 5, to prove that if a sequence E = {e 1,e2 , ... } has the property that En DI- 0 for every sequence A of positive upper density 6 then lim inf e.+1/e. = I. • 1 1 1-+co Theorem 8 and the above conjecture are related to a general problem of Motzkin who asked how dense a sequence A can be if V(A) does not contain any elements from a given set K. Cantor and Gordon [I] and more recently Haralambis [4], have obtained some results in this connexion, mainly for finite sets K. Sarkozy [JO], [II] and [12] considered the case of some interesting infinite sets K....
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