Packing Chromatic Number of Certain Graphs

William, Albert; Roy, Snigdha

doi:10.12732/ijpam.v87i6.1

Cited by 10 publications

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References 12 publications

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“…The question of whether the packing chromatic number of (sub)cubic graphs is bounded was open for several years. The answer is positive for several families of such graphs, for instance for (all subgraphs of) the infinite 3-regular tree (see Section 2.2), for (all subgraphs of) the infinite hexagonal lattice (see Section 6.1), for Sierpiński-type graphs (see Section 5.2), and for the so-called H-graphs [19,52,68]. Additional exact values or the upper bounds of the packing chromatic numbers of some additional (sub)cubic graphs can be found in [7,35,50,52].…”

Section: Subcubic Graphsmentioning

confidence: 99%

“…667]. William and Roy [68,Proposition 6] proved that if n ≥ 8, then χ ρ (P n K 1 ) ≤ 5. Laïche, Bouchemakh, and Sopena extended this result as follows.…”

Section: Corona Graphs and Specific Classesmentioning

confidence: 99%

“…William, Roy, and Rajasingh in [67] investigated, among others, kC 2n -linear graphs, gear graphs, barbell graphs, tadpole graphs, and lollipop graphs. William and Roy [68] further studied the packing chromatic number of windmill graphs and generalized theta graphs. Tight bounds for the packing chromatic number of the latter graphs were derived by Laïche, Bouchemakh and Sopena [51].…”

Section: Corona Graphs and Specific Classesmentioning

confidence: 99%

See 2 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 Graphsmentioning

confidence: 99%

“…667]. William and Roy [68,Proposition 6] proved that if n ≥ 8, then χ ρ (P n K 1 ) ≤ 5. Laïche, Bouchemakh, and Sopena extended this result as follows.…”

Section: Corona Graphs and Specific Classesmentioning

confidence: 99%

Section: Corona Graphs and Specific Classesmentioning

confidence: 99%

See 1 more Smart Citation

A survey on packing colorings

Brešar¹,

Ferme²,

Klavžar³

et al. 2020

Discuss. Math. Graph Theory

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“…A packing k-coloring of a graph G is a mapping π : V(G) → {1, 2,...,k} such that any two vertices of color i are at least i + 1. The packing chromatic number χ p (G) of G is the smallest integer k for which G has packing k-coloring [2].…”

Section: Introductionmentioning

confidence: 99%

Packing chromatic number of transformation graphs

Durgun

Ozen-Dortok

2019

THERM SCI

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“…Thus, the vertices of G are partitioned into different color classes X 1 , X 2 , ..., X k , where every X i is an i-packing of G. The i-packing number of G, denoted by ρ i (G), is the maximum cardinality of an i-packing that occurs in G. The packing chromatic number χ p (G) of G is the smallest integer k for which G has packing k-coloring [3,4,10,14].…”

Section: Introductionmentioning

confidence: 99%

Some bounds on local connective Chromatic number

Dündar¹,

Çiftçi²

2017

ntmsci

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Graph coloring is one of the most important concept in graph theory. Many practical problems can be formulated as graph coloring problems. In this paper, we define a new coloring concept called local connective coloring. A local connective k-coloring of a graph G is a proper vertex coloring, which assigns colors from {1, 2, ..., k} to the vertices V (G) in a such way that any two nonadjacent vertices u and v of a color i satisfies κ(u, v) i, where κ(u, v) is the maximum number of internally disjoint paths between u and v. Adjacent vertices are colored with different colors as in the proper coloring. The smallest integer k for which there exists a local connective k-coloring of G is called the local connective chromatic number of G, and it is denoted by χ lc (G). We study this coloring on several classes of graphs and give some general bounds. We also compare local connective chromatic number of a graph with chromatic number and packing chromatic number of it.

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Packing Chromatic Number of Certain Graphs

Cited by 10 publications

References 12 publications

A survey on packing colorings

A survey on packing colorings

Packing chromatic number of transformation graphs

Some bounds on local connective Chromatic number

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