The packing coloring problem for lobsters and partner limited graphs

Argiroffo, Gabriela R.; Nasini, Graciela L.; Torres, Pablo Daniel

doi:10.1016/j.dam.2012.08.008

Cited by 22 publications

(12 citation statements)

References 5 publications

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“…Fiala and Golovach have shown that the decision version of the packing chromatic number is NP-complete even in the class of trees [12]. Packing coloring of some other classes of graphs, such as the distance graphs [11,22,25], hypercubes [26], subdivision graphs of subcubic graphs [4,10,15], and still other classes of graphs [2,18,20] was also studied.…”

Section: Introductionmentioning

confidence: 99%

Packing colorings of subcubic outerplanar graphs

2020

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Given a graph G and a nondecreasing sequence S = (s1, . . . , s k ) of positive integers, the mapping c : V (G) −→ {1, . . . , k} is called an Spacking coloring of G if for any two distinct vertices x and y in c −1 (i), the distance between x and y is greater than si. The smallest integer k such that there exists a (1, 2, . . . , k)-packing coloring of a graph G is called the packing chromatic number of G, denoted χρ(G). The question of boundedness of the packing chromatic number in the class of subcubic (planar) graphs was investigated in several earlier papers; recently it was established that the invariant is unbounded in the class of all subcubic graphs.In this paper, we prove that the packing chromatic number of any 2-connected bipartite subcubic outerplanar graph is bounded by 7. Furthermore, we prove that every subcubic triangle-free outerplanar graph has a (1, 2, 2, 2)-packing coloring, and that there exists a subcubic outerplanar graph with a triangle that does not admit a (1, 2, 2, 2)-packing coloring. In addition, there exists a subcubic triangle-free outerplanar graph that does not admit a (1, 2, 2, 3)-packing coloring. A similar dichotomy is shown for bipartite outerplanar graphs: every such graph admits an Spacking coloring for S = (1, 3, . . . , 3), where 3 appears ∆ times (∆ being the maximum degree of vertices), and if one of the integers 3 is replaced by 4 in the sequence S. Also, there exist outerplanar bipartite graphs that do not admit an S-packing coloring.

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Section: Introductionmentioning

confidence: 99%

Packing colorings of subcubic outerplanar graphs

2020

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“…The concept has attracted a considerable attention recently: there are around 30 papers on the topic (see e.g. [1,3,4,5,6,7,8,9,10,11,12,13,22] and references in them). In particular, Fiala and Golovach [10] proved that finding the packing chromatic number of a graph is NP-hard even in the class of trees.…”

Section: Introductionmentioning

confidence: 99%

Packing Chromatic Number of Subdivisions of Cubic Graphs

Balogh

Kostochka

Liu

2019

Graphs and Combinatorics

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“…Indeed, the packing chromatic number is intrinsically more difficult due to the fact that determining χ ρ is NP-complete even when restricted to trees [7]. On the other hand, Argiroffo et al discovered that the packing coloring problem is solvable in polynomial time for several nontrivial classes of graphs [2]. In addition, the packing chromatic number was studied on hypercubes [10,16], Cartesian product graphs [11,13], and distance graphs [6,15].…”

Section: Introductionmentioning

confidence: 99%

Packing chromatic number under local changes in a graph

Brear

Klavar

Rall

et al. 2017

Discrete Mathematics

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The packing chromatic number χ ρ (G) of a graph G is the smallest integer k such that there exists a k-vertex coloring of G in which any two vertices receiving color i are at distance at least i + 1. It is proved that in the class of subcubic graphs the packing chromatic number is bigger than 13, thus answering an open problem from [Gastineau, Togni, S-packing colorings of cubic graphs, Discrete Math. 339 (2016) 2461-2470]. In addition, the packing chromatic number is investigated with respect to several local operations. In particular, if S e (G) is the graph obtained from a graph G by subdividing its edge e, then ⌊χ ρ (G)/2⌋ + 1 ≤ χ ρ (S e (G)) ≤ χ ρ (G) + 1.

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The packing coloring problem for lobsters and partner limited graphs

Cited by 22 publications

References 5 publications

Packing colorings of subcubic outerplanar graphs

Packing colorings of subcubic outerplanar graphs

Packing Chromatic Number of Subdivisions of Cubic Graphs

Packing chromatic number under local changes in a graph

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