Connectivity and diameter in distance graphs

Penso, Lúcia Draque; Rautenbach, Dieter; Szwarcfiter, Jayme Luiz

doi:10.1002/net.20397

Cited by 9 publications

(4 citation statements)

References 19 publications

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“…For infinitely large N , these multiloop networks become graphs which are called distance graphs. In fact, circulant graphs coincide exactly with the regular distance graphs as it was shown in [18]. Distance graphs were introduced by Eggleton et al in [6], a lot of papers on different kinds of colorings of distance graphs have been published in last 20 years, including some results for L(2, 1)-labelings (see [10,20,21]).…”

Section: Introductionmentioning

confidence: 93%

Radio Labelings of Distance Graphs

Čada,

Ekstein,

Holub

et al. 2012

Preprint

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Motivated by the Channel Assignment Problem, we study radio k-labelings of graphs. A radio k-labeling of a connected graph G is an assignment c of non negative integers to the vertices of G such thatfor any two vertices x and y, x = y, where d(x, y) is the distance between x and y in G.In this paper, we study radio k-labelings of distance graphs, i.e., graphs with the set Z of integers as vertex set and in which two distinct vertices i, j ∈ Z are adjacent if and only if |i − j| ∈ D. We give some lower and upper bounds for radio k-labelings of distance graphs with distance sets D(1, 2, . . . , t), D(1, t) and D(t − 1, t) for any positive integer t > 1.

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Section: Introductionmentioning

confidence: 93%

Radio Labelings of Distance Graphs

Čada,

Ekstein,

Holub

et al. 2012

Preprint

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“…In fact, circulant graphs coincide exactly with the regular distance graphs [43]. In [14], [15], [43], [44], [45], some fundamental results concerning circulant graphs are extended to the more general class of distance graphs.…”

Section: Circulant Graphs In the Study Of Distance Graphsmentioning

confidence: 99%

Vertex Coloring of Certain Distance Graphs

Yegnanarayanan¹,

Parthiban²

2013

Int. J. of Pure and Appl. Math.

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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.

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“…More specifically, distance graphs are the subgraphs of sufficiently large circulant graphs induced by sets of consecutive vertices. Conversely, our following simple observation from [13] shows that every circulant graph is in fact a distance graph.…”

Section: Introductionmentioning

confidence: 96%

Long cycles and paths in distance graphs

Penso

Rautenbach

Szwarcfiter

2010

Discrete Mathematics

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For n ∈ N and D ⊆ N, the distance graph P D n has vertex set {0, 1,. .. , n − 1} and edge set {ij | 0 ≤ i, j ≤ n − 1, |j − i| ∈ D}. Note that the important and very well-studied circulant graphs coincide with the regular distance graphs. A fundamental result concerning circulant graphs is that for these graphs, a simple greatest common divisor condition, their connectivity, and the existence of a Hamiltonian cycle are all equivalent. Our main result suitably extends this equivalence to distance graphs. We prove that for a set D, there is a constant c D such that the greatest common divisor of the integers in D is 1 if and only if for every n, P D n has a component of order at least n − c D if and only if for every n, P D n has a cycle of order at least n − c D. Furthermore, we discuss some consequences and variants of this result.

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Connectivity and diameter in distance graphs

Cited by 9 publications

References 19 publications

Radio Labelings of Distance Graphs

Radio Labelings of Distance Graphs

Vertex Coloring of Certain Distance Graphs

Long cycles and paths in distance graphs

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