Subdivision into i-packings and S-packing chromatic number of some lattices

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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“…. , k)-packing coloring has been the subject of many papers (see [7,8,13]) since its introduction by Goddard et al [18].…”

Section: Introductionmentioning

confidence: 99%

On S-packing edge-colorings of cubic graphs

Gastineau

Togni

2019

Discrete Applied Mathematics

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“…Note that the range 13-15 for (d, n) = (1, 1) is from Theorem 6.2, while the values for (n, d) ∈ {(2, 4), (2, 5)} follow from Proposition 7.42. The remaining values in regular font are from [31], those in bold font are improvements derived in [46], and further improvements from [48] Tables parallel to Table 4 for H, T , P ∞ P ∞ , P 2 P ∞ , and the octagonal grid were also produced by Gastineau, Kheddouci and Togni [31], and improved by Korže and Vesel [46].…”

Section: Triangular and Hexagonal Latticesmentioning

confidence: 99%

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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“…Furthermore, if 1 a / ∈ f 2 (N (w 4 ) − w 1 ) and 2 a / ∈ f 2 (N (w 3 ) − w 1 ), then we can recolor w 1 with 2 a and color w and w 4 with 1 a . With (15) and (18), all vertices in Figure 4). Then we can color w with 3 a .…”

Section: Proof Of Theoremmentioning

confidence: 99%