Integral distance graphs

Chen, Jer-Jeong; Chang, Gerard J.; Huang, Kuo‐Ching

doi:10.1002/(sici)1097-0118(199708)25:4<287::aid-jgt6>3.0.co;2-g

Cited by 40 publications

(11 citation statements)

References 13 publications

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“…We get similar contradictions if v and w are both negative or v is positive and w is negative, and these cases are handled in the appendix. When v and w are both negative and t 3, we will only need to use inequalities (5) and (6) to arrive at a contradiction. When v and w are negative and t = 1 or 2, we also use the inequality derived from Lemma 7 to arrive at a contradiction.…”

Section: Proof Of Theoremmentioning

confidence: 98%

“…A difference graph is an undirected Cayley graph of the group (Z, +) with generators being the elements of S. Every vertex of G(S) has degree 2|S|, and in particular, (G(S)) = 2|S|. We will need a folklore lemma on the chromatic number of difference graphs due to Chen, Chang, and Huang [6], which we prove for completeness. Lemma 2 (Chen et al [6]).…”

Section: Coloring Lemmasmentioning

confidence: 99%

“…We will need a folklore lemma on the chromatic number of difference graphs due to Chen, Chang, and Huang [6], which we prove for completeness. Lemma 2 (Chen et al [6]). For all subsets S ⊂ N, we have…”

Section: Coloring Lemmasmentioning

confidence: 99%

“…When v and w are negative and t = 1 or 2, we also use the inequality derived from Lemma 7 to arrive at a contradiction. When v is positive and w is negative, we only need to use the inequalities (5) and (6) to arrive at a contradiction when t 2. When v is positive and w is negative and t = 1, we can use the inequalities derived from congruences (3) and (4) to arrive at a contradiction.…”

Section: Proof Of Theoremmentioning

confidence: 99%

See 3 more Smart Citations

On Rado's Boundedness Conjecture

Fox¹,

Kleitman²

2006

Journal of Combinatorial Theory, Series A

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We prove that Rado's Boundedness Conjecture from Richard Rado's 1933 famous dissertation Studien zur Kombinatorik is true if it is true for homogeneous equations. We then prove the first nontrivial case of Rado's Boundedness Conjecture: if a 1 , a 2 , and a 3 are integers, and if for every 24-coloring of the positive integers (or even the nonzero rational numbers) there is a monochromatic solution to the equation a 1 x 1 + a 2 x 2 + a 3 x 3 = 0, then for every finite coloring of the positive integers there is a monochromatic solution to a 1 x 1 + a 2 x 2 + a 3 x 3 = 0.

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Section: Proof Of Theoremmentioning

confidence: 98%

Section: Coloring Lemmasmentioning

confidence: 99%

Section: Coloring Lemmasmentioning

confidence: 99%

Section: Proof Of Theoremmentioning

confidence: 99%

See 2 more Smart Citations

On Rado's Boundedness Conjecture

Fox¹,

Kleitman²

2006

Journal of Combinatorial Theory, Series A

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“…Otherwise, χ(D) = 3. After some partial results [2,3,4,9] the case |D| = 3 was finally settled by Xuding Zhu in [12], where sufficiently accurate bounds on the circular and fractional chromatic numbers of G(D) are obtained to prove the following result. In this paper Zhu conjectured that, if the upper bound χ(G(D)) = |D| + 1 is achieved then D contains three not necessarily distinct elements a, b, c with a + b = c. This conjecture was further strengthened to give the following one:…”

Section: Introductionmentioning

confidence: 99%

Distance graphs with maximum chromatic number

Barajas

Serra

2008

Discrete Mathematics

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Let D be a finite set of integers. The distance graph G(D) has the set of integers as vertices and two vertices at distance d ∈ D are adjacent in G(D). A conjecture of Xuding Zhu states that if the chromatic number of G(D) achieves its maximum value |D| + 1 then the graph has a clique of order |D|. We prove that the chromatic number of a distance graph with D = {a, b, c, d} is five if and only if either D = {1, 2, 3, 4k} or D = {a, b, a + b, a + 2b} with a ≡ 0 (mod 2) and b ≡ 1 (mod 2). This confirms Zhu's conjecture for |D| = 4.

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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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Integral distance graphs

Cited by 40 publications

References 13 publications

On Rado's Boundedness Conjecture

On Rado's Boundedness Conjecture

Distance graphs with maximum chromatic number

Distance graphs with missing multiples in the distance sets

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