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Journal ArticleDOI

On a question concerning prime distance graphs

28 Feb 2002-Discrete Mathematics (Elsevier Science Publishers B. V.)-Vol. 245, Iss: 1, pp 293-298
TL;DR: A conditional answer to this question based on a well-known conjecure from the prime number theory is given.
About: This article is published in Discrete Mathematics.The article was published on 2002-02-28 and is currently open access. It has received 7 citations till now. The article focuses on the topics: Twin prime & Almost prime.
Citations
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Journal ArticleDOI
TL;DR: It is proved that trees, cycles, and bipartite graphs are prime distance graphs, and that Dutch windmill graphs and paper mill graphs arePrime distance graphs if and only if the Twin Prime Conjecture and dePolignac’s Conjectures are true, respectively.

8 citations


Cites background from "On a question concerning prime dist..."

  • ...Research in prime distance graphs has since focused on the chromatic number of Z(D) where D is a non-empty proper subset of P [7, 6, 17, 19]....

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Journal ArticleDOI
TL;DR: In this article, the authors considered the problem of characterizing class three and class four sets when the distance set D is not only a subset of primes P but also a special class of prime numbers like Additive and Deletable primes, Wedderburn-Etherington Number primes and Highly Cototient number primes.
Abstract: An integer distance graph is a graph G(Z,D) with the set of integers as vertex set and an edge joining two vertices u and v if and only if ∣u - v∣D where D is a subset of the positive integers. It is known that x(G(Z,D) )=4 where P is a set of Prime numbers. So we can allocate the subsets D of P to four classes, accordingly as is 1 or 2 or 3 or 4. In this paper we have considered the open problem of characterizing class three and class four sets when the distance set D is not only a subset of primes P but also a special class of primes like Additive primes, Deletable primes, Wedderburn-Etherington Number primes, Euclid-Mullin sequence primes, Motzkin primes, Catalan primes, Schroder primes, Non-generous primes, Pell primes, Primeval primes, Primes of Binary Quadratic Form, Smarandache-Wellin primes, and Highly Cototient number primes. We also have indicated the membership of a number of special classes of prime numbers in class 2 category.

6 citations

Journal ArticleDOI
TL;DR: In this paper, an integer distance graph is defined as a graph G(Z,D) with the set of integers as vertex set and an edge joining two vertices u and v if and only if |u − v| ∈ D where D is a subset of the positive integers.
Abstract: In this paper first, we give a brief introduction about integer distance graphs. An integer distance graph is a graph G(Z,D) with the set of integers as vertex set and an edge joining two vertices u and v if and only if |u − v| ∈ D where D is a subset of the positive integers. If D is a subset of P then we call G(Z,D) a prime distance graph. Second, we obtain a partial solution to a general open problem of characterizing a class of prime distance graphs. Third, we compute the vertex arboricity of certain prime distance graphs. Fourth, we give a brief review regarding circulant graphs and highlight its importance in the computation of chromatic number of distance graphs with appropriate references. Fifth, we introduce the notion of pseudochromatic coloring and obtain certain results concerning circulant graphs and distance graphs.

2 citations

Posted Content
TL;DR: A 2-odd graph is a graph whose vertices can be labeled with distinct integers such that for any two adjacent vertices, the difference of their labels is prime (either 2 or odd) as discussed by the authors.
Abstract: A graph $G$ is a prime distance graph (respectively, a 2-odd graph) if its vertices can be labeled with distinct integers such that for any two adjacent vertices, the difference of their labels is prime (either 2 or odd). We prove that trees, cycles, and bipartite graphs are prime distance graphs, and that Dutch windmill graphs and paper mill graphs are prime distance graphs if and only if the Twin Prime Conjecture and dePolignac's Conjecture are true, respectively. We give a characterization of 2-odd graphs in terms of edge colorings, and we use this characterization to determine which circulant graphs of the form $Circ(n, \{1,k\})$ are 2-odd and to prove results on circulant prime distance graphs.
Posted Content
TL;DR: In this article, it was shown that a graph is a strict prime if and only if its vertices can be labeled with distinct integers such that for any two adjacent vertices, the difference of their labels is the product of at most two primes.
Abstract: A graph $G$ is a $k$-prime product distance graph if its vertices can be labeled with distinct integers such that for any two adjacent vertices, the difference of their labels is the product of at most $k$ primes. A graph has prime product number $ppn(G)=k$ if it is a $k$-prime product graph but not a $(k-1)$-prime product graph. Similarly, $G$ is a prime $k$th-power graph (respectively, strict prime $k$th-power graph) if its vertices can be labeled with distinct integers such that for any two adjacent vertices, the difference of their labels is the $j$th power of a prime, for $j \leq k$ (respectively, the $k$th power of a prime exactly). We prove that $ppn(K_n) = \lceil \log_2(n)\rceil - 1$, and for a nonempty $k$-chromatic graph $G$, $ppn(G) = \lceil \log_2(k)\rceil - 1$ or $ppn(G) = \lceil \log_2(k)\rceil$. We determine $ppn(G)$ for all complete bipartite, 3-partite, and 4-partite graphs. We prove that $K_n$ is a prime $k$th-power graph if and only if $n < 7$, and we determine conditions on cycles and outerplanar graphs $G$ for which $G$ is a strict prime $k$th-power graph. We find connections between prime product and prime power distance graphs and the Twin Prime Conjecture, the Green-Tao Theorem, and Fermat's Last Theorem.
References
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Book
17 Dec 1994
TL;DR: In this article, the Conjectures of Hadwiger and Hajos are used to define graph types, such as planar graph, graph on higher surfaces, and critical graph.
Abstract: Planar Graphs. Graphs on Higher Surfaces. Degrees. Critical Graphs. The Conjectures of Hadwiger and Hajos. Sparse Graphs. Perfect Graphs. Geometric and Combinatorial Graphs. Algorithms. Constructions. Edge Colorings. Orientations and Flows. Chromatic Polynomials. Hypergraphs. Infinite Chromatic Graphs. Miscellaneous Problems. Indexes.

1,380 citations


"On a question concerning prime dist..." refers background or methods in this paper

  • ...For more on these methods the readers are referred to [8,9,3,5,19]....

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  • ...For distance graphs on the real line R or the integer set Z , the problem of 3nding the chromatic numbers of G(R;D) or G(Z; D) for diAerent sets D has been studied extensively [16,17,3,5]....

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Journal ArticleDOI
TL;DR: I was asked to write a paper about the major unsolved problems in com-binatorial mathematics, but after some thought it seemed better to modify the title to a less pretentious one, so I state only my three favourite problems.
Abstract: I was asked to write a paper about the major unsolved problems in com-binatorial mathematics. After some thought it seemed better to modify the title to a less pretentious one. Combinatorial mathematics has grown enormously and a genuine survey would have to include not only topics where I have no real competence but also topics about which I n e v er seriously thought, e.g. algorithmic combinatorics, coding theory and matroid theory. There is no doubt that the proof of the conjecture that several simply stated problems have no good algorithm is fundamental and may h a ve i m-portant consequences for many other branches of mathematics, but unfortunately I have no real feeling for these questions and I feel I should leave the subject to those who are more competent. I just heard that Khachiyan 59], has a polynomial algorithm for linear programming. (See also 50].) This is considered a sensational result and during my last stay i n the U.S. many o f m y friends were greatly impressed by i t. First of all I will discuss some problems on set systems. I state only my three favourite problems, but before starting I refer to the survey paper 31].

299 citations


"On a question concerning prime dist..." refers background in this paper

  • ...Erdős [10] mentioned this problem as one of his favorites....

    [...]

Book
01 Jan 1988
TL;DR: In this article, the authors present a list of primality tests for natural numbers and prove that a natural number is a prime in the sense that it can be seen as a function of the number of prime numbers.
Abstract: 1. How Many Prime Numbers Are There?.- I. Euclid's Proof.- II. Kummer's Proof.- III. Polya's Proof.- IV. Euler's Proof.- V. Thue's Proof.- VI. Two-and-a-Half Forgotten Proofs.- A. Perott's Proof.- B. Auric's Proof.- C. Metrod's Proof.- VII. Washington's Proof.- VIII. Furstenberg's Proof.- 2. How to Recognize Whether a Natural Number Is a Prime?.- I. The Sieve of Eratosthenes.- II. Some Fundamental Theorems on Congruences.- A. Fermat's Little Theorem and Primitive Roots Modulo a Prime.- B. The Theorem of Wilson.- C. The Properties of Giuga, Wolstenholme and Mann & Shanks.- D. The Power of a Prime Dividing a Factorial.- E. The Chinese Remainder Theorem.- F. Euler's Function.- G. Sequences of Binomials 31.- H. Quadratic Residues.- III. Classical Primality Tests Based on Congruences.- IV. Lucas Sequences.- Addendum on Lehmer Numbers.- V. Classical Primality Tests Based on Lucas Sequences.- VI. Fermat Numbers.- VII. Mersenne Numbers.- Addendum on Perfect Numbers.- VIII. Pseudoprimes.- A. Pseudoprimes in Base 2 (psp).- B. Pseudoprimes in Base a (psp(a)).- B. Euler Pseudoprimes in Base a (epsp(a)).- D. Strong Pseudoprimes in Base a (spsp(a)).- Addendum on the Congruence an-k ? bn-k (mod n).- Carmichael Numbers.- Addendum on Knodel Numbers.- X. Lucas Pseudoprimes.- A. Fibonacci Pseudoprimes.- B. Lucas Pseudoprimes (e psp(P, Q)).- C. Euler-Lucas Pseudoprimes (e e psp(P, Q)) and Strong Lucas Pseudoprimes (s psp(P, Q)).- D. Carmichael-Lucas Numbers.- XI. Last Section on Primality Testing and Factorization!.- A. The Cost of Testing.- B. Recent Primality Tests.- C. Monte Carlo Methods.- D. Titanic Primes.- E. Factorization.- F. Public Key Cryptography.- 3. Are There Functions Defining Prime Numbers?.- I. Functions Satisfying Condition (a).- II. Functions Satisfying Condition (b).- III. Functions Satisfying Condition (c).- 4. How Are the Prime Numbers Distributed?.- I. The Growth of ?(x).- A. History Unfolding.- Euler.- Legendre.- Gauss.- Tschebycheff.- Riemann.- de la Vallee Poussin and Hadamard.- Erdos and Selberg.- B. Sums Involving the Mobius Function.- C. The Distribution of Values of Euler's Function.- D. Tables of Primes.- E. The Exact Value of ?(x) and Comparison with x/(log x), Li(x), and R(x).- F. The Nontrivial Zeroes of ?(s).- G. Zero-Free Regions for ?(s) and the Error Term in the Prime Number Theorem.- H. The Growth of ?(s).- II. The nth Prime and Gaps.- A. Some Properties of ?(x).- B. The nth Prime.- C. Gaps between Primes.- D. The Possible Gaps between Primes.- E. Interlude.- III. Twin Primes.- Addendum on Polignac's Conjecture.- IV. Primes in Arithmetic Progression.- A. There Are Infinitely Many!.- B. The Smallest Prime in an Arithmetic Progression.- C. Strings of Primes in Arithmetic Progression.- V. Primes in Special Sequences.- VI. Goldbach's Famous Conjecture.- VII. The Waring-Goldbach Problem.- A. Waring's Problem.- B. The Waring-Goldbach Problem.- VIII. The Distribution of Pseudoprimes and of Carmichael Numbers.- A. Distribution of Pseudoprimes.- B. Distribution of Carmichael Numbers.- C. Distribution of Lucas Pseudoprimes.- 5. Which Special Kinds of Primes Have Been Considered?.- I. Regular Primes.- II. Sophie Germain Primes.- III. Wieferich Primes.- IV. Wilson Primes.- V. Repunits and Similar Numbers.- VI. Primes with Given Initial and Final Digits.- VII. Numbers k x 2' +- 1.- Addendum on Cullen Numbers.- VIII. Primes and Second-Order Linear Recurrence Sequences.- IX. The NSW-Primes.- 6. Heuristic and Probabilistic Results About Prime Numbers.- I. Prime Values of Linear Polynomials.- II. Prime Values of Polynomials of Arbitrary Degree.- III. Some Probabilistic Estimates.- A. Partitio Numerorum.- B. Polynomials with Many Successive Composite Values.- C. Distribution of Mersenne Primes.- D. The log log Philosophy.- IV. The Density of the Set of Regular Primes.- Conclusion.- Dear Reader:.- Citations for Some Possible Prizes for Work on the Prime Number Theorem.- A. General References.- B. Specific References.- 1.- 2.- 3.- 4.- 5.- 6.- Conclusion.- Primes up to 10,000.- Index of Names.- Gallimawfries.

299 citations

Journal ArticleDOI
01 Jan 1951
TL;DR: In this paper, the Cartesian product of a family of compact sets is shown to be compact, based on TychonofI's theorem, which is a special case of R. RADO's theorem.
Abstract: Our original proof was simplified by SZEKERES. Later, a simple proof, based on TychonofI’s theorem that the Cartesian product of a family of compact sets is compact, was indicated by RABSOX and A. STONE. We suppress these proofs here, since t’heorem 1 can be considered as a special case of a theorem of R. RADO which appeared meanwhile [3], and a topological proof for Rado’s theorem was given by GOTTSCHALK [2].

290 citations


"On a question concerning prime dist..." refers result in this paper

  • ...Erdős [10] mentioned this problem as one of his favorites....

    [...]

  • ...This was proved by de Bruijn and Erdős [6]....

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  • ...This was proved by de Bruijn and Erdős [6]....

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