Rainbow neighbourhood number of graphs
Abstract In this paper, we introduce the notion of the rainbow neighbourhood and a related graph parameter namely the rainbow neighbourhood number and report on preliminary results thereof. The closed neighbourhood N [v] of a vertex v ∈ V (G) which contains at least one coloured vertex o...
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| Lenguaje: | English |
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Universidad Católica del Norte, Departamento de Matemáticas
2019
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oai:scielo:S0716-091720190003004692019-08-28Rainbow neighbourhood number of graphsKok,JohanNaduvath,SudevJamil,Muhammad Kamran Colour cluster Colour classes Rainbow neighbourhood Expanded line graph v-clique 05C07 05C38 05C75 05C85 Abstract In this paper, we introduce the notion of the rainbow neighbourhood and a related graph parameter namely the rainbow neighbourhood number and report on preliminary results thereof. The closed neighbourhood N [v] of a vertex v ∈ V (G) which contains at least one coloured vertex of each colour in the chromatic colouring of a graph is called a rainbow neighbourhood. The number of rainbow neighbourhoods in a graph G is called the rainbow neighbourhood number of G, denoted by rχ(G). We also introduce the concepts of an expanded line graph of a graph G and a v-clique of v ∈ V (G). With the help of these new concepts, we also establish a necessary and sufficient condition for the existence of a rainbow neighbourhood in the line graph of a graph G.info:eu-repo/semantics/openAccessUniversidad Católica del Norte, Departamento de MatemáticasProyecciones (Antofagasta) v.38 n.3 20192019-08-01text/htmlhttp://www.scielo.cl/scielo.php?script=sci_arttext&pid=S0716-09172019000300469en10.22199/issn.0717-6279-2019-03-0030 |
| institution |
Scielo Chile |
| collection |
Scielo Chile |
| language |
English |
| topic |
Colour cluster Colour classes Rainbow neighbourhood Expanded line graph v-clique 05C07 05C38 05C75 05C85 |
| spellingShingle |
Colour cluster Colour classes Rainbow neighbourhood Expanded line graph v-clique 05C07 05C38 05C75 05C85 Kok,Johan Naduvath,Sudev Jamil,Muhammad Kamran Rainbow neighbourhood number of graphs |
| description |
Abstract In this paper, we introduce the notion of the rainbow neighbourhood and a related graph parameter namely the rainbow neighbourhood number and report on preliminary results thereof. The closed neighbourhood N [v] of a vertex v ∈ V (G) which contains at least one coloured vertex of each colour in the chromatic colouring of a graph is called a rainbow neighbourhood. The number of rainbow neighbourhoods in a graph G is called the rainbow neighbourhood number of G, denoted by rχ(G). We also introduce the concepts of an expanded line graph of a graph G and a v-clique of v ∈ V (G). With the help of these new concepts, we also establish a necessary and sufficient condition for the existence of a rainbow neighbourhood in the line graph of a graph G. |
| author |
Kok,Johan Naduvath,Sudev Jamil,Muhammad Kamran |
| author_facet |
Kok,Johan Naduvath,Sudev Jamil,Muhammad Kamran |
| author_sort |
Kok,Johan |
| title |
Rainbow neighbourhood number of graphs |
| title_short |
Rainbow neighbourhood number of graphs |
| title_full |
Rainbow neighbourhood number of graphs |
| title_fullStr |
Rainbow neighbourhood number of graphs |
| title_full_unstemmed |
Rainbow neighbourhood number of graphs |
| title_sort |
rainbow neighbourhood number of graphs |
| publisher |
Universidad Católica del Norte, Departamento de Matemáticas |
| publishDate |
2019 |
| url |
http://www.scielo.cl/scielo.php?script=sci_arttext&pid=S0716-09172019000300469 |
| work_keys_str_mv |
AT kokjohan rainbowneighbourhoodnumberofgraphs AT naduvathsudev rainbowneighbourhoodnumberofgraphs AT jamilmuhammadkamran rainbowneighbourhoodnumberofgraphs |
| _version_ |
1718439851302322176 |