The packing chromatic number of infinite product graphs

Fiala, Jiřı́; Klavar, Sandi; Lidický, Bernard

doi:10.1016/j.ejc.2008.09.014

Cited by 54 publications

(53 citation statements)

References 2 publications

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“…4. h It was showed in [10] that v q ðGÃZÞ < 1 for any finite graph G. It follows that v q ðP n ÃP m ÃZÞ < 1 for any m; n. Some results on the packing chromatic number of P 2 ÃP 3 ÃP k are presented in [12], while the proposed integer linear programming models and satisfiability test model enables the computation for some larger families of graphs. The results are presented in Table 5.…”

Section: Packing Chromatic Number Of C M ãC N and P M ãC Nmentioning

confidence: 96%

“…The upper bound was improved to 17 by Holub and Soukal [8], while Ekstein et al [9] showed that a packing 11-coloring of the square lattice cannot exist. Some finite and infinite subgraphs of the square lattice as well as some related graphs, in particular the Cartesian products of paths and cycles, have been also extensively explored [3,[9][10][11]8,12].…”

Section: Introductionmentioning

confidence: 99%

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Modeling the packing coloring problem of graphs

Shao

Vesel

2015

Applied Mathematical Modelling

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a b s t r a c tThe frequency assignment problem asks for assigning frequencies to transmitters in a wireless network and includes a variety of specific subproblems. In particular, the packing coloring problem comes from the regulations concerning the assignment of broadcast frequencies to radio stations. The problem is placed on a graph-theoretical footing and modeled as the packing chromatic number v q ðGÞ of a graph G. The packing chromatic number of G is the smallest integer k such that the vertex set VðGÞ can be partitioned into disjoint classes X 1 ; . . . ; X k , with the condition that vertices in X i have pairwise distance greater than i. The determination of the packing chromatic number is known to be NP-hard. This paper presents an integer linear programming model and a satisfiability test model for the packing coloring problem. The proposed models were applied for studying the packing chromatic numbers of Cartesian products of paths and cycles and for the packing chromatic numbers of some distance graphs.

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Section: Packing Chromatic Number Of C M ãC N and P M ãC Nmentioning

confidence: 96%

Section: Introductionmentioning

confidence: 99%

Modeling the packing coloring problem of graphs

Shao

Vesel

2015

Applied Mathematical Modelling

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“…Upper bounds on the size of X i are useful in establishing lower bounds on χ S (G). Density was formally introduced by Fiala et al [6]. Definition 1 ([6]).…”

Section: Densitymentioning

confidence: 99%

“…There have been several papers on the packing chromatic number, especially on the values for lattices and other infinite graphs [1][2][3][6][7][8][12][13][14], as well as on the computational complexity of the parameter [5,8].…”

Section: Introductionmentioning

confidence: 99%

A note onS-packing colorings of lattices

Goddard

2014

Discrete Applied Mathematics

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a b s t r a c tLet a 1 , a 2 , . . . , a k be positive integers. An (a 1 , a 2 , . . . , a k )-packing coloring of a graph G is a mapping from V (G) to {1, 2, . . . , k} such that vertices with color i have pairwise distance greater than a i . In this paper, we study (a 1 , a 2 , . . . , a k )-packing colorings of several lattices including the infinite square, triangular, and hexagonal lattices. For k small, we determine all a i such that these graphs have packing colorings. We also give some exact values and asymptotic bounds.

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“…Much work has been devoted to the packing chromatic number of graphs ( [2], [3], [4], [5], [6], [7], [8]). Fiala and Golovach [5] showed that determining the packing chromatic number is an NP-hard problem for trees.…”

Section: Introductionmentioning

confidence: 99%

The packing chromatic number of hypercubes

Torres

Valencia-Pabon²

2015

Discrete Applied Mathematics

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The packing chromatic number χ ρ (G) of a graph G is the smallest integer k needed to proper color the vertices of G in such a way that the distance in G between any two vertices having color i be at least i + 1. Goddard et al. [8] found an upper bound for the packing chromatic number of hypercubes Q n . Moreover, they compute χ ρ (Q n ) for n ≤ 5 leaving as an open problem the remaining cases. In this paper, we obtain a better upper bound for χ ρ (Q n ) and we compute the exact value of χ ρ (Q n ) for 6 ≤ n ≤ 8.

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The packing chromatic number of infinite product graphs

Cited by 54 publications

References 2 publications

Modeling the packing coloring problem of graphs

Modeling the packing coloring problem of graphs

A note onS-packing colorings of lattices

The packing chromatic number of hypercubes

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