On the packing chromatic number of subcubic outerplanar graphs

Gastineau, Nicolas; Holub, Přemysl; Togni, Olivier

doi:10.1016/j.dam.2018.07.034

Cited by 23 publications

(15 citation statements)

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“…The vertex analogous of S-packing edge-coloring has been first studied by Goddard and Xu [16,17] and then recently on cubic graphs [2,3,4,12,14]. The particular case of (1, 2, .…”

Section: Introductionmentioning

confidence: 99%

On S-packing edge-colorings of cubic graphs

Gastineau

Togni

2019

Discrete Applied Mathematics

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Given a non-decreasing sequence S = (s 1 , s 2 , . . . , s k ) of positive integers, an Spacking edge-coloring of a graph G is a partition of the edge set of G into k subsets {X 1 , X 2 , . . . , X k } such that for each 1 ≤ i ≤ k, the distance between two distinct edges e, e ′ ∈ X i is at least s i + 1. This paper studies S-packing edgecolorings of cubic graphs. Among other results, we prove that cubic graphs having a 2-factor are (1, 1, 1, 3, 3)-packing edge-colorable, (1, 1, 1, 4, 4, 4, 4, 4)-packing edgecolorable and (1, 1, 2, 2, 2, 2, 2)-packing edge-colorable. We determine sharper results for cubic graphs of bounded oddness and 3-edge-colorable cubic graphs and we propose many open problems.

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“…The vertex analogous of S-packing edge-coloring has been first studied by Goddard and Xu [16,17] and then recently on cubic graphs [2,3,4,12,14]. The particular case of (1, 2, .…”

Section: Introductionmentioning

confidence: 99%

On S-packing edge-colorings of cubic graphs

Gastineau

Togni

2019

Discrete Applied Mathematics

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“…, 3k − 1)-packing coloring. The next two results were proved by Gastineau, Holub, and Togni [30] in their investigation of packing colorings of outerplanar graphs. Suppose c is an (a 1 , a 2 , . .…”

Section: Infinite Pathsmentioning

confidence: 75%

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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“…Similarly, by (18), we may assume f 2 (N (w 6 ) − w 2 ) = {1 a , 1 b } (See Figure 3). With (14), (17), and the case, 3 a / ∈ f 2 (B(w, 3) − {w}) and we can extend f 2 to G by assigning f 2 (w) = 3 a . Case 2.1.2:…”

Section: Proof Of Theoremmentioning

confidence: 99%