The packing chromatic number of hypercubes

Torres, Pablo Daniel; Valencia-Pabon, Mario

doi:10.1016/j.dam.2015.04.006

Cited by 21 publications

(2 citation statements)

References 13 publications

(6 reference statements)

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“…✷ Consider the hypercubes Q n , n ≥ 1. Indeed, χ ρ (Q n ) was determined exactly for n ≤ 5 in Goddard et al (2008) and for 6 ≤ n ≤ 8 in Torres and Valencia-Pabon (2015). Moreover, χ ρ (Q n ) asymptotically grows as…”

Section: Jacobs Et Al (2013)mentioning

confidence: 96%

Packing chromatic vertex-critical graphs

Klavžar,

Rall

2018

Preprint

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The packing chromatic number χρ(G) of a graph G is the smallest integer k such that the vertex set of G can be partitioned into sets Vi, i ∈ [k], where vertices in Vi are pairwise at distance at least i + 1. Packing chromatic vertex-critical graphs, χρ-critical for short, are introduced as the graphs G for which χρ(G − x) < χρ(G) holds for every vertex} can be almost arbitrary. The 3-χρ-critical graphs are characterized, and 4-χρcritical graphs are characterized in the case when they contain a cycle of length at least 5 which is not congruent to 0 modulo 4. It is shown that for every integer k ≥ 2 there exists a k-χρ-critical tree and that a k-χρ-critical caterpillar exists if and only if k ≤ 7. Cartesian products are also considered and in particular it is proved that if G and H are vertex-transitive graphs and diam(G) + diam(H) ≤ χρ(G), then G H is χρ-critical.

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Section: Jacobs Et Al (2013)mentioning

confidence: 96%

Packing chromatic vertex-critical graphs

Klavžar,

Rall

2018

Preprint

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“…Using the fact that Q k = Q k−1 K 2 and applying Corollary 5.2, Torres and Valencia-Pabon [65] proved the following explicit lower bound and upper bound for the packing chromatic number of hypercubes, respectively.…”

Section: Products Of Complete Graphsmentioning

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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Packing Chromatic Number of Windmill Related Graphs and Chain Silicate Networks

Augustine,

Santiago

2023

Trends in Mathematics

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The packing chromatic number of hypercubes

Cited by 21 publications

References 13 publications

Packing chromatic vertex-critical graphs

Packing chromatic vertex-critical graphs

A survey on packing colorings

Packing Chromatic Number of Windmill Related Graphs and Chain Silicate Networks

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