On Graph Fall-Coloring: Existence and Constructions

Kaul, Hemanshu; Mitillos, Christodoulos

doi:10.1007/s00373-019-02082-7

Cited by 5 publications

(2 citation statements)

References 10 publications

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“…If G is an arbitrary connected and not idomatic graph, then the theorem implies that IC( G) = 0. For instance, a cycle C n is not idomatic if and only if n is odd and n is not congruent modulo 3, see [14]. Additional families of graphs that are not idomatic were constructed in [6], for instance graphs G with χ(G) > δ(G) + 1.…”

Section: Graphs G With Ic(g) ∈ {0 N(g) N(g) − 1}mentioning

confidence: 99%

On independent coalition in graphs and independent coalition graphs

Alikhani,

Bakhshesh,

Golmohammadi

et al. 2024

Discuss. Math. Graph Theory

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An independent coalition in a graph G consists of two disjoint, indepen-either the set V i consists of a single dominating vertex of G, or V i forms an independent coalition with some other part V j . The independent coalition number IC(G) of G is the maximum order of an independent coalition of G. The independent coalition graph ICG(G, π) of π = {V 1 , . . . , V k } (and of G) has the vertex set {V 1 , . . . , V k }, vertices V i and V j being adjacent if V i and V j form an independent coalition in G. In this paper, a large family of graphs with IC(G) = 0 is described and graphs

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Section: Graphs G With Ic(g) ∈ {0 N(g) N(g) − 1}mentioning

confidence: 99%

On independent coalition in graphs and independent coalition graphs

Alikhani,

Bakhshesh,

Golmohammadi

et al. 2024

Discuss. Math. Graph Theory

View full text Add to dashboard Cite

show abstract

“…The problem of b-chromatic numbers was introduced by Irving and Manlove in 1999 [1] and studied extensively in the literature (see the survey in [2]), whereas fall coloring was introduced in [3] and studied in [4][5][6]. It follows from [6] that fall coloring strongly chordal graphs is doable in polynomial time, even with an unbounded number of colors.…”

Section: Introductionmentioning

confidence: 99%

On Fall-Colorable Graphs

Wang,

Wen,

Wang

et al. 2024

Mathematics

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A fall k-coloring of a graph G is a proper k-coloring of G such that each vertex has at least one neighbor in each of the other color classes. A graph G which has a fall k-coloring is equivalent to having a partition of the vertex set V(G) in k independent dominating sets. In this paper, we first prove that for any fall k-colorable graph G with order n, the number of edges of G is at least (n(k−1)+r(k−r))/2, where r≡n(modk) and 0≤r≤k−1, and the bound is tight. Then, we obtain that if G is k-colorable (k≥2) and the minimum degree of G is at least k−2k−1n, then G is fall k-colorable and this condition of minimum degree is the best possible. Moreover, we give a simple proof for an NP-hard result of determining whether a graph is fall k-colorable, where k≥3. Finally, we show that there exist an infinite family of fall k-colorable planar graphs for k∈{5,6}.

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Complexity of Fall Coloring for Restricted Graph Classes

Lauri

Mitillos

2020

Theory Comput Syst

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On Graph Fall-Coloring: Existence and Constructions

Cited by 5 publications

References 10 publications

On independent coalition in graphs and independent coalition graphs

On independent coalition in graphs and independent coalition graphs

On Fall-Colorable Graphs

Complexity of Fall Coloring for Restricted Graph Classes

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