Dominated Colorings of Graphs

Merouane, Houcine Boumediene; Haddad, Mohammed; Chellali, Mustapha; Kheddouci, Hamamache

doi:10.1007/s00373-014-1407-3

Cited by 34 publications

(26 citation statements)

References 10 publications

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“…Moreover, if G is not a complete graph and its complement, then there are two vertices in G with distance 2 and they are assigned the same color. Therefore we have: Observation 1.3 [10] If G is a triangle-free graph, then χ dom (G) ≤ 2γ(G) . Proposition 1.4 [10] Let G be a graph with order at least 2.…”

Section: Introductionmentioning

confidence: 99%

“…Dominated coloring of a graph is a proper coloring in which each color class is dominated by a vertex. The least number of colors needed for a dominated coloring of G is called the dominated chromatic number of G and is denoted by χ dom (G) [10]. We call this coloring a dom-coloring, for simplicity.…”

Section: Introductionmentioning

confidence: 99%

“…Observation 1.3 [10] If G is a triangle-free graph, then χ dom (G) ≤ 2γ(G) . Proposition 1.4 [10] Let G be a graph with order at least 2. Then χ dom (G) ≥ γ t (G) .…”

Section: Introductionmentioning

confidence: 99%

“…Moreover, if G is a triangle-free graph, then χ dom (G) = γ t (G). Proposition 1.5 [10] Let G be a graph without isolated vertices. Then χ dom (G) ≤ χ(G).γ(G).…”

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

On dominated coloring of graphs and some Nordhaus–Gaddum-type relations

Choopani¹,

Jafarzadeh²,

Erfanian³

et al. 2018

Turk J Math

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The dominated coloring of a graph G is a proper coloring of G such that each color class is dominated by at least one vertex. The minimum number of colors needed for a dominated coloring of G is called the dominated chromatic number of G , denoted by χ dom (G). In this paper, dominated coloring of graphs is compared with (open) packing number of G and it is shown that if G is a graph of order n with diam(G) ≥ 3 , then χ dom (G) ≤ n − ρ(G) and if ρ0(G) = 2n/3 , then χ dom (G) = ρ0(G) , and if ρ(G) = n/2 , then χ dom (G) = ρ(G). The dominated chromatic numbers of the corona of two graphs are investigated and it is shown that if µ(G) is the Mycielsky graph of G , then we have χ dom (µ(G)) = χ dom (G) + 1. It is also proved that the Vizing-type conjecture holds for dominated colorings of the direct product of two graphs. Finally we obtain some Nordhaus-Gaddum-type results for the dominated chromatic number χ dom (G) .

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Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

“…Observation 1.3 [10] If G is a triangle-free graph, then χ dom (G) ≤ 2γ(G) . Proposition 1.4 [10] Let G be a graph with order at least 2. Then χ dom (G) ≥ γ t (G) .…”

Section: Introductionmentioning

confidence: 99%

“…Moreover, if G is a triangle-free graph, then χ dom (G) = γ t (G). Proposition 1.5 [10] Let G be a graph without isolated vertices. Then χ dom (G) ≤ χ(G).γ(G).…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

On dominated coloring of graphs and some Nordhaus–Gaddum-type relations

Choopani¹,

Jafarzadeh²,

Erfanian³

et al. 2018

Turk J Math

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show abstract

Section: Introduction and Definitionsmentioning

confidence: 99%

Introduction to dominated edge chromatic number of a graph

Piri¹,

Alikhani²

2020

Preprint

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We introduce and study the dominated edge coloring of a graph. A dominated edge coloring of a graph G is a proper edge coloring of G such that each color class is dominated by at least one edge of G. The minimum number of colors among all dominated edge coloring is called the dominated edge chromatic number, denoted by χ ′ dom (G). We obtain some properties of χ ′ dom (G) and compute it for specific graphs. Also we examine the effects on χ ′ dom (G) when G is modified by operations on vertex and edge of G. Finally, we consider the k-subdivision of G and study the dominated edge chromatic number of these kind of graphs.

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On cd-Coloring of Trees and Co-bipartite Graphs

Shalu

Kirubakaran

2021

Algorithms and Discrete Applied Mathematics

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Dominated Colorings of Graphs

Cited by 34 publications

References 10 publications

On dominated coloring of graphs and some Nordhaus–Gaddum-type relations

On dominated coloring of graphs and some Nordhaus–Gaddum-type relations

Introduction to dominated edge chromatic number of a graph

On cd-Coloring of Trees and Co-bipartite Graphs

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