Packing Chromatic Number of Subdivisions of Cubic Graphs

Balogh, József; Kostochka, Alexandr; Liu, Xujun

doi:10.1007/s00373-019-02016-3

Cited by 25 publications

(35 citation statements)

References 26 publications

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“…. , 3,4) as in the statement of the proposition. Note that there exists a vertex x ∈ {r} ∪ N (r) that is colored by 1.…”

Section: Proof Of Theoremmentioning

confidence: 94%

“…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%

“…. , 3,4), where 3 appears ∆(G) − 1 times, there exists a bipartite outerplanar graph that does not admit an S-packing coloring.…”

Section: Introductionmentioning

confidence: 99%

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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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“…. , 3,4) as in the statement of the proposition. Note that there exists a vertex x ∈ {r} ∪ N (r) that is colored by 1.…”

Section: Proof Of Theoremmentioning

confidence: 94%

Section: Introductionmentioning

confidence: 99%

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Packing colorings of subcubic outerplanar graphs

2020

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“…From this, one can infer that in the class of graphs with degree bounded by k the packing chromatic number is unbounded as soon as k ≥ 4. Several recent papers considered the question of whether the packing chromatic number in the class of graphs with maximum degree 3 (i.e., subcubic graphs) is bounded by some constant [5,6,15], but in the very recent, not yet published paper [2] the authors prove that this is not the case. Nevertheless, the question of boundedness of the packing chromatic number in some natural infinite classes of graphs remains interesting.…”

Section: Introductionmentioning

confidence: 99%

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confidence: 99%

Packing coloring of Sierpiński-type graphs

Brešar

Ferme

2018

Aequat. Math.

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The packing chromatic number χ ρ (G) of a graph G is the smallest integer k such that the vertex set of G can be partitioned into sets V i , i ∈ {1, . . . , k}, where each V i is an i-packing. In this paper, we consider the packing chromatic number of several families of Sierpiński-type graphs. While it is known that this number is bounded from above by 8 in the family of Sierpiński graphs with base 3, we prove that it is unbounded in the families of Sierpiński graphs with bases greater than 3. On the other hand, we prove that the packing chromatic number in the family of Sierpiński triangle graphs ST n 3 is bounded from above by 31. Furthermore, we establish or provide bounds for the packing chromatic numbers of generalized Sierpiński graphs S n G with respect to all connected graphs G of order 4.

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Every subcubic multigraph is (1,27) $(1,{2}^{7})$‐packing edge‐colorable

Liu

Santana

Short

2023

Journal of Graph Theory

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For a nondecreasing sequence of positive integers, an ‐packing edge‐coloring of a graph is a decomposition of edges of into disjoint sets such that for each the distance between any two distinct edges is at least . The notion of ‐packing edge‐coloring was first generalized by Gastineau and Togni from its vertex counterpart. They showed that there are subcubic graphs that are not ‐packing (abbreviated to ‐packing) edge‐colorable and asked the question whether every subcubic graph is ‐packing edge‐colorable. Very recently, Hocquard, Lajou, and Lužar showed that every subcubic graph is ‐packing edge‐colorable and every 3‐edge‐colorable subcubic graph is ‐packing edge‐colorable. Furthermore, they also conjectured that every subcubic graph is ‐packing edge‐colorable.In this paper, we confirm the conjecture of Hocquard, Lajou, and Lužar, and extend it to multigraphs.

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Packing Chromatic Number of Subdivisions of Cubic Graphs

Cited by 25 publications

References 26 publications

Packing colorings of subcubic outerplanar graphs

Packing colorings of subcubic outerplanar graphs

Packing coloring of Sierpiński-type graphs

Every subcubic multigraph is (1,27) $(1,{2}^{7})$‐packing edge‐colorable

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