<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si24.gif" display="inline" overflow="scroll"><mml:mi>S</mml:mi></mml:math>-packing colorings of cubic graphs

Gastineau, Nicolas; Togni, Olivier

doi:10.1016/j.disc.2016.04.017

Cited by 34 publications

(46 citation statements)

References 14 publications

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“…Cranston and Kim [16,Theorem 1] proved that if G is a subcubic graph and G is not the Petersen graph, then G is (2, 2, 2, 2, 2, 2, 2, 2)-packing colorable. Gastineau and Togni studied S-packing colorings of subdivisions of graphs [32]. The following result of theirs provides a connection between S-packing colorings of a graph G and those of the subdivision graph S(G), and, as we will see, it suggests a method of attacking Conjecture 4.14.…”

Section: Subcubic and Subdivided Graphsmentioning

confidence: 81%

“…For a long time it was not known whether there are subcubic graphs with a large packing chromatic number. Gastineau and Togni [32,Proposition 6] provided an example of a cubic graph of order 38, whose packing chromatic number is 13 (and also an example of a cubic graph of order 24 and packing chromatic number 11). They also posed the question of whether there exists a (sub)cubic graph with larger packing chromatic number.…”

Section: Subcubic Graphsmentioning

confidence: 99%

“…Gastineau and Togni [32] asked whether the subdivision of any subcubic graph is packing 5-colorable, while Brešar, Klavžar, Rall, and Wash [14] turned the question into a conjecture. Since χ ρ (S(G)) = 3 for any connected bipartite graph G of order at least 3, and every subcubic graph is a subgraph of a cubic graph, it suffices to consider only cubic non-bipartite graphs.…”

Section: Edge Contraction and Edge Subdivisionmentioning

confidence: 99%

“…Theorem 7.24 [32,Theorem 2]. If G is a 3-irregular, subcubic graph, then G is (1, 2, 2, 2)-packing colorable.…”

Section: Subcubic and Subdivided Graphsmentioning

confidence: 99%

“…Proposition 7.25 [32,Corollary 2]. The following statements are equivalent for every graph G with δ(G) ≥ 3:…”

Section: Subcubic and Subdivided Graphsmentioning

confidence: 99%

See 4 more Smart Citations

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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Section: Subcubic and Subdivided Graphsmentioning

confidence: 81%

Section: Subcubic Graphsmentioning

confidence: 99%

Section: Edge Contraction and Edge Subdivisionmentioning

confidence: 99%

“…Theorem 7.24 [32,Theorem 2]. If G is a 3-irregular, subcubic graph, then G is (1, 2, 2, 2)-packing colorable.…”

Section: Subcubic and Subdivided Graphsmentioning

confidence: 99%

“…Proposition 7.25 [32,Corollary 2]. The following statements are equivalent for every graph G with δ(G) ≥ 3:…”

Section: Subcubic and Subdivided Graphsmentioning

confidence: 99%

See 3 more Smart Citations

A survey on packing colorings

Brešar¹,

Ferme²,

Klavžar³

et al. 2020

Discuss. Math. Graph Theory

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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 Windmill Related Graphs and Chain Silicate Networks

Augustine,

Santiago

2023

Trends in Mathematics

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S-packing colorings of cubic graphs

Cited by 34 publications

References 14 publications

A survey on packing colorings

A survey on packing colorings

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

Packing Chromatic Number of Windmill Related Graphs and Chain Silicate Networks

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