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

Jacobs, Yolandé; Jonck, Elizabeth; Joubert, Ernst J.

doi:10.2478/s11533-013-0237-5

Cited by 7 publications

(7 citation statements)

References 3 publications

(1 reference statement)

Supporting

Mentioning

Contrasting

Order By: Relevance

“…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%

Modeling the packing coloring problem of graphs

Shao

Vesel

2015

Applied Mathematical Modelling

View full text Add to dashboard Cite

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.

show abstract

Section: Introductionmentioning

confidence: 99%

Modeling the packing coloring problem of graphs

Shao

Vesel

2015

Applied Mathematical Modelling

View full text Add to dashboard Cite

show abstract

“…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

View full text Add to dashboard Cite

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.

show abstract

“…The packing chromatic number of the Cartesian product was already considered in [1] where the general upper and lower bound were set. The lower bound was later improved in [8]. Several exact values and bounds for special families of Cartesian product graphs can be found in [8,9].…”

Section: Introductionmentioning

confidence: 99%

“…The lower bound was later improved in [8]. Several exact values and bounds for special families of Cartesian product graphs can be found in [8,9].…”

Section: Introductionmentioning

confidence: 99%

A note on the packing chromatic number of lexicographic products

Božović¹,

Peterin²

2019

Preprint

View full text Add to dashboard Cite

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. In this short note we present upper and lower bound for the packing chromatic number of the lexicographic product G • H of graphs G and H. Both bounds coincide in many cases. In particular this happens if |V (H)| − α(H) ≥ diam(G) − 1, where α(G) denotes the independence number of G.

show abstract

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

Cited by 7 publications

References 3 publications

Modeling the packing coloring problem of graphs

Modeling the packing coloring problem of graphs

Packing colorings of subcubic outerplanar graphs

A note on the packing chromatic number of lexicographic products

Contact Info

Product

Resources

About