On the packing chromatic number of Cartesian products, hexagonal lattice, and trees

Brešar, Boštjan; Klavar, Sandi; Rall, Douglas F.

doi:10.1016/j.dam.2007.06.008

Cited by 87 publications

(78 citation statements)

References 5 publications

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“…By given a different color greater than 2⌊ n−4 2 ⌋ + 3 to the remaining vertices in Q n , we obtain the desired packing coloring of Q n with 3 + 2 n ( 1 2 − 1 2 ⌈log 2 n⌉ ) − 2⌊ n−4 2 ⌋ colors. 4 3 Lower bounds for χ ρ (Q n ) : the cases n = 6, 7 and 8…”

Section: Proof Of Theoremmentioning

confidence: 99%

“…The notion of packing chromatic number was established by Goddard et al [8] under the name broadcast chromatic 1 number. The term packing chromatic number was introduced by Brešar et al [4].…”

Section: Introductionmentioning

confidence: 99%

“…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%

See 2 more Smart Citations

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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Section: Proof Of Theoremmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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The packing chromatic number of hypercubes

Torres

Valencia-Pabon²

2015

Discrete Applied Mathematics

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“…This colouring was first introduced by Goddard et al [7] where it was called broadcast colouring. Brešar et al [1] were the first to use the term packing colouring.…”

Section: Introductionmentioning

confidence: 99%

A lower bound for the packing chromatic number of the Cartesian product of cycles

2013

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Let G = (V E) be a simple graph of order and be an integer with ≥ 1. The set X ⊆ V (G) is called an -packing if each two distinct vertices in X are more than apart. A packing colouring of G is a partition X = {X 1 X 2 X } of V (G) such that each colour class X is an -packing. The minimum order of a packing colouring is called the packing chromatic number of G, denoted by χ ρ (G). In this paper we show, using a theoretical proof, that if = 4 , for some integer ≥ 3, then 9 ≤ χ ρ (C 4 2C ) ≤ 11. We will also show that if is a multiple of four, then χ ρ (C 4 2C ) = 9.MSC: 05C69

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“…In 2008, Goddard et al [14] establised the notion of packing chromatic number under the name broadcast chromatic number. The term packing chromatic number was introduced by Brešar et al [4]. The concept of packing coloring comes from the area of frequency planning in wireless networks.…”

Section: Introductionmentioning

confidence: 99%

Some bounds on local connective Chromatic number

Dündar¹,

Çiftçi²

2017

ntmsci

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Graph coloring is one of the most important concept in graph theory. Many practical problems can be formulated as graph coloring problems. In this paper, we define a new coloring concept called local connective coloring. A local connective k-coloring of a graph G is a proper vertex coloring, which assigns colors from {1, 2, ..., k} to the vertices V (G) in a such way that any two nonadjacent vertices u and v of a color i satisfies κ(u, v) i, where κ(u, v) is the maximum number of internally disjoint paths between u and v. Adjacent vertices are colored with different colors as in the proper coloring. The smallest integer k for which there exists a local connective k-coloring of G is called the local connective chromatic number of G, and it is denoted by χ lc (G). We study this coloring on several classes of graphs and give some general bounds. We also compare local connective chromatic number of a graph with chromatic number and packing chromatic number of it.

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On the packing chromatic number of Cartesian products, hexagonal lattice, and trees

Cited by 87 publications

References 5 publications

The packing chromatic number of hypercubes

The packing chromatic number of hypercubes

A lower bound for the packing chromatic number of the Cartesian product of cycles

Some bounds on local connective Chromatic number

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