Some New Results on the Rainbow Neighbourhood Number of Graphs

Naduvath, Sudev; Susanth, C; Kalayathankal, Sunny Joseph; Kok, Johan

doi:10.1007/s40009-018-0740-0

Cited by 9 publications

(7 citation statements)

References 1 publication

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“…The rainbow neighbourhood number of certain fundamental graph classes have been determined in [4,6,7]. Some of the relevant results in these paper are as follows:…”

Section: Rainbow Neighbourhoods In a Graphmentioning

confidence: 99%

On the Rainbow Neighbourhood Number of Mycielski Type Graphs

Naduvath¹,

Susanth²,

Kalayathankal³

2019

Int. J. of Appl. Math.

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A rainbow neighbourhood of a graph G is the closed neighbourhood N [v] of a vertex v ∈ V (G) which contains at least one colored vertex of each color in the chromatic coloring C of G. Let G be a graph with a chromatic coloring C defined on it. The number of vertices in G yielding rainbow neighbourhoods is called the rainbow neighbourhood number of the graph G, denoted by r χ (G). In this paper, we discuss the rainbow neighbourhood number of the Mycielski type graphs of graphs.

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“…The rainbow neighbourhood number of certain fundamental graph classes have been determined in [4,6,7]. Some of the relevant results in these paper are as follows:…”

Section: Rainbow Neighbourhoods In a Graphmentioning

confidence: 99%

On the Rainbow Neighbourhood Number of Mycielski Type Graphs

Naduvath¹,

Susanth²,

Kalayathankal³

2019

Int. J. of Appl. Math.

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“…A review of results known to the authors shows that thus far, the minimum rainbow neighbourhood number is implicitly defined [3,4,5,6,7]. It is proposed that henceforth the notation r − χ (G) replaces r χ (G) and that r χ (G) only refers to the number of rainbow neighbourhoods found for any given minimum proper colouring allocation.…”

Section: Maximum Rainbow Neighbourhood Number R + χ (G) Permitted By ...mentioning

confidence: 99%

“…For the main part of this paper the notation remains as found in the literature [3,4,5,6,7]. Later we introduce an appropriate change in subsection 2.1.…”

Section: Introductionmentioning

confidence: 99%

Reflection on rainbow neighbourhood numbers of graphs

Kok,

Naduvath,

Buelban

2017

Preprint

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A rainbow neighbourhood of a graph G with respect to a proper colouringconsists of vertices from all colour classes in G with respect to C. The number of vertices in G which yield a rainbow neighbourhood of G is called its rainbow neighbourhood number. In this paper, we show that all results known so far about the rainbow neighbourhood number of a graph G implicitly refer to a minimum number of vertices which yield rainbow neighbourhoods in respect of the minimum proper colouring where the colours are allocated in accordance with the rainbow neighbourhood convention. Relaxing the aforesaid convention allows for determining a maximum rainbow neighbourhood number of a graph G. We also establish the fact that the minimum and maximum rainbow neighbourhood numbers are respectively, unique and therefore a constant for a given graph.

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“…Note that the closed neighbourhood [ ] of a vertex ∈ ( ) which contains at least one vertex from each colour class of in the chromatic colouring, is called a rainbow neighbourhood (see [4][5][6][7] for further results on rainbow neighbourhoods of different graphs). The number of vertices in which yield rainbow neighbourhoods, denoted by ( ), is called the rainbow neighbourhood number of corresponding to the chromatic colouring.…”

Section: Introductionmentioning

confidence: 99%

Generalisation of the rainbow neighbourhood number and 𝑘-jump colouring of a graph

Kok¹,

Naduvath²,

Mphako-Banda³

2020

AMI

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In this paper, the notions of rainbow neighbourhood and rainbow neighbourhood number of a graph are generalised and further to these generalisations, the notion of a proper-jump colouring of a graph is also introduced. The generalisations follow from the understanding that a closedneighbourhood of a vertex ∈ () denoted, [ ] is the set, [ ] = { : (,) ≤ , ∈ N and ≤ ()}. If the closed-neighbourhood [ ] contains at least one of each colour of the chromatic colour set, we say that yields a-rainbow neighbourhood.

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Some New Results on the Rainbow Neighbourhood Number of Graphs

Cited by 9 publications

References 1 publication

On the Rainbow Neighbourhood Number of Mycielski Type Graphs

On the Rainbow Neighbourhood Number of Mycielski Type Graphs

Reflection on rainbow neighbourhood numbers of graphs

Generalisation of the rainbow neighbourhood number and 𝑘-jump colouring of a graph

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