On the independence ratio of distance graphs

Carraher, James M.; Galvin, David; Hartke, Stephen G.; Radcliffe, A. J.; Stolee, Derrick

doi:10.1016/j.disc.2016.05.018

Cited by 7 publications

(14 citation statements)

References 39 publications

(38 reference statements)

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“…In this study, we relate our results with the already known results for the families M = {1, 2, x, x + 2} [14] and M = {2, 3, x, x + 2} [10]. Our Corollary 2.6 for a = 1, provides the counterexamples to the conjecture [2,Conjecture 29]. Although, different counterexamples to this conjecture were already provided by Liu and Robinson [15].…”

Section: Introductionsupporting

confidence: 57%

“…Collister and Liu [10] has discussed κ(M ) and µ(M ) for M = {2, 3, x, y} with |x − y| ≤ 6. Very recently, Liu and Robinson [15,Theorems 1,2,3] have found µ(M ) and κ(M ) and proved their equality for most of the remaining cases of the family {1, x, y}. Their results also provide the counterexamples for the two conjectures of Carraher et al [2].…”

Section: Introductionmentioning

confidence: 88%

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Maximal Density of Integral Sets with Missing Differences and the Kappa Values

Pandey

Srivastava

2022

Taiwanese J. Math.

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Let M be a given set of positive integers. A set S of nonnegative integers is said to be an M -set if a, b ∈ S implies a−b / ∈ M . In an unpublished problem collection, Motzkin asked to find maximal upper asymptotic density, denoted by µ(M ), of Msets. The first published work on µ(M ) is due to Cantor and Gordon in 1973, in which, they found the exact value of µ(M ) when |M | ≤ 2. In fact, this is the only general case, in which, we have a closed formulae for µ(M ). If |M | ≥ 3, then the exact value of µ(M ) is not known for the general set M . In the past six decades or so, several attempts have been given to study µ(M ) but µ(M ) has been found exactly or estimated only in very few cases. In this paper, we study µ(M ) for the families M = {a, a + 1, x} and M = {a, a + 1, x, y}, where y − x ≤ 2 and y > x > a + 1. Our results in the case of M = {a, a + 1, x} also give counterexamples to a conjecture of Carraher. Although, different counterexamples to this conjecture, were already given by Liu and Robinson in 2020. We also relate our results with the already know results for the families M = {1, 2, x, x + 2} and M = {2, 3, x, x + 2}.

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Section: Introductionsupporting

confidence: 57%

Section: Introductionmentioning

confidence: 88%

Maximal Density of Integral Sets with Missing Differences and the Kappa Values

Pandey

Srivastava

2022

Taiwanese J. Math.

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“…Motzkin's problem also has connections with some other problems, such as problems related to the fractional chromatic number of distance graphs and the Lonely Runner Conjecture. One can refer to [9,[20][21][22][23][24][25][26][27][28][29].…”

Section: Introductionmentioning

confidence: 99%

On Optimal M-Sets Related to Motzkin’s Problem

Yang

Pan

2020

Journal of Mathematics

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Let M be a set of positive integers. A set S of nonnegative integers is called an M ‐ set if a and b ∈ S , then a − b ∉ M . If S ⊆ 0,1 , … , n is an M − set with the maximal cardinality, then S is called a maximal M − set of 0,1 , … , n . If S ∩ 0,1 , … , n is a maximal M − set of 0,1 , … , n for all integers n ≥ 0 , then we call S an optimal M − set. In this paper, we study the existence of an optimal M − set.

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“…On the one hand, integer distance graphs are natural generalizations of circulant graphs. In fact, the chromatic number of an undirected integer distance graph Γ(Z, ±S), where ±S := {±s : s ∈ S}, has been extensively studied before; see, e.g., Carraher, Galvin, Hartke, Radcliffe, and Stolee [1]. On the other hand, an integer distance graph can be viewed as the limit of a sequence of circulant graphs and understanding domination in integer distance graphs may shed light on the asymptotic behavior of domination in large circulant graphs.…”

Section: Introductionmentioning

confidence: 99%

“…Replacing limit inferior with limit superior in (1.1) gives the upper density of U ⊆ Z. Carraher, Galvin, Hartke, Radcliffe, and Stolee [1] used upper density to study independent sets in an undirected integer distance graph Γ(Z, ±S). We provide some results on lower density in Section 2, with similar proofs to previous work [1]. For example, the following result proved in Section 2 is a natural extension of an analogous result on independence ratio [1,Theorem 4].…”

Section: Introductionmentioning

confidence: 99%

Domination ratio of integer distance digraphs

Huang

2019

Preprint

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An integer distance digraph is the Cayley graph Γ(Z, S) of the additive group Z of all integers with respect to some finite subset S ⊆ Z. The domination ratio of Γ(Z, S) is the minimum density of a dominating set in Γ(Z, S). We establish some basic results on the domination ratio of Γ(Z, S) and precisely determine it when S = {s, t} with s dividing t. Recently there has been some research work on the existence of an efficient dominating set in a finite Cayley graph; see, e.g., and Dejter-Serra [5]. An efficient dominating set, also called a perfect code, of a digraph Γ is a dominating set D such that every vertex of Γ is dominated by exactly one vertex in D. For a finite Cayley graph Γ = Γ(G, S) with n = |G| vertices, each having d = |S| outgoing edges, there is a straightforward lower bound γ(Γ) ≥ n/(1 + d), where the equality holds if and only if there exists an efficient dominating set.A circulant (di)graph is a finite Cayley graph Γ(Z n , S) where Z n is the finite cyclic group of integers modulo n and S is a subset of Z n . When −S := {−s : s ∈ S} coincides with S, the digraph Γ(Z, S) can be viewed as an undirected graph. Circulant graphs provide important topological structures for interconnection networks due to their symmetry, fault-tolerance, routing capabilities, and other good properties, and have been used in telecommunication networks, VLSI design, and distributed computation. Domination in circulant graphs has been studied by Huang-Xu [7], Kumar-MacGillivray [8], Obradović-Peters-Ružić [11], Rad [12], and others. Let γ(Z n , S) denote the domination number of Γ(Z n , S). The following results are known for the double loop network Γ(Z n , {1, s})

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On the independence ratio of distance graphs

Cited by 7 publications

References 39 publications

Maximal Density of Integral Sets with Missing Differences and the Kappa Values

Maximal Density of Integral Sets with Missing Differences and the Kappa Values

On Optimal M-Sets Related to Motzkin’s Problem

Domination ratio of integer distance digraphs

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