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Ekstein, Jan; Holub, Přemysl; Togni, Olivier

doi:10.1016/j.dam.2013.10.036

Cited by 24 publications

(13 citation statements)

References 13 publications

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“…The study of a packing coloring of distance graphs Dðk; tÞ was initiated by Togni [13] who determined the packing chromatic number of Dð1; tÞ for small values of t. Distance graphs were further investigated by Ekstein et al in [14,15], where some more results on the packing chromatic number of Dð1; tÞ and Dðk; tÞ are given.…”

Section: Introductionmentioning

confidence: 99%

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: Introductionmentioning

confidence: 99%

Modeling the packing coloring problem of graphs

Shao

Vesel

2015

Applied Mathematical Modelling

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“…Some of these bounds were improved by Ekstein, Holub, and Lidický [22]; the improved bounds are presented in the same table in bold font. Next, Ekstein, Holub, and Togni [23] further improved two values; these are the underlined items in the table. Finally, Shao and Vesel [60] improved three of the upper bounds; these improvements are marked with stars in the table.…”

Section: Distance Graphsmentioning

confidence: 99%

“…This was the key observation in proving the next result. Ekstein et al [23] further improved the bounds on the packing chromatic number of some specific distance graphs as follows. The particular case when k = 1 of Theorem 6.10 was earlier deduced in [22,Corollary 4].…”

Section: Distance Graphsmentioning

confidence: 99%

See 1 more Smart Citation

A survey on packing colorings

Brešar¹,

Ferme²,

Klavžar³

et al. 2020

Discuss. Math. Graph Theory

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If S = (a 1 , a 2 ,. . .) is a non-decreasing sequence of positive integers, then an S-packing coloring of a graph G is a partition of V (G) into sets X 1 , X 2 ,. .. such that for each pair of distinct vertices in the set X i , the distance between them is larger than a i. If there exists an integer k such that V (G) = X 1 ∪ • • • ∪ X k , then the partition is called an S-packing k-coloring. The S-packing chromatic number of G is the smallest k such that G admits an S-packing k-coloring. If a i = i for every i, then the terminology reduces to packing colorings and packing chromatic number. Since the introduction of these generalizations of the chromatic number in 2008 more than fifty papers followed. Here we survey the state of the art on the packing coloring, and its generalization, the S-packing coloring. We also list several conjectures and open problems.

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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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The packing coloring of distance graphs D(k,t)

Cited by 24 publications

References 13 publications

Modeling the packing coloring problem of graphs

Modeling the packing coloring problem of graphs

A survey on packing colorings

Packing colorings of subcubic outerplanar graphs

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