<abstract xmlns="http://www.w3.org/1999/xhtml">A set <italic>S</italic> ⊆ <italic>V</italic> is a neighborhood set of a graph <italic>G</italic> = (<italic>V, E</italic>), if <italic>G</italic> = ∪<italic>v∈S</italic> 〈 <italic>N</italic>[<italic>v</italic>] 〉, where 〈 <italic>N</italic>[<italic>v</italic>] 〉 is the subgraph of a graph <italic>G</italic> induced by <italic>v</italic> and all vertices adjacent to <italic>v</italic>. A neighborhood set <italic>S</italic> is said to be a neighbor coloring set if it contains at least one vertex from each color class of a graph <italic>G</italic>, where color class of a colored graph is the set of vertices having one particular color. The neighbor chromatic number χ<italic>n</italic> (<italic>G</italic>) is the minimum cardinality of a neighbor coloring set of a graph <italic>G</italic>. In this article, some results on neighbor chromatic number of Cartesian products of two paths (grid graph) and cycles (torus graphs) are explored.</abstract>

A set S ⊆ V is a neighborhood set of a graph G = (V, E), if G = ∪v∈S 〈 N[v] 〉, where 〈 N[v] 〉 is the subgraph of a graph G induced by v and all vertices adjacent to v. A neighborhood set S is said to be a neighbor coloring set if it contains at least one vertex from each color class of a graph G, where color class of a colored graph is the set of vertices having one particular color. The neighbor chromatic number χn (G) is the minimum cardinality of a neighbor coloring set of a graph G. In this article, some results on neighbor chromatic number of Cartesian products of two paths (grid graph) and cycles (torus graphs) are explored.

On Neighbor chromatic number of grid and torus graphs

General Mathematics

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{"article-title":"On Neighbor chromatic number of grid and torus graphs"}

A set S ⊆ V is a neighborhood set of a graph G = (V, E), if G = ∪v∈S 〈 N[v] 〉, where 〈 N[v] 〉 is the subgraph of a graph G induced by v and all vertices...

On Neighbor chromatic number of grid and torus graphs

Published Online: Dec 21, 2019

Page range: 3 - 15

Received: Apr 25, 2018

Accepted: Jun 17, 2018

DOI: https://doi.org/10.2478/gm-2019-0001

Keywords
Neighborhood number, chromatic number, neighbor chromatic number, grid graph, torus graph

© 2019 B. Chaluvaraju et al., published by Sciendo

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

On Neighbor chromatic number of grid and torus graphs

Published Online: Dec 21, 2019

Page range: 3 - 15

Received: Apr 25, 2018

Accepted: Jun 17, 2018

DOI: https://doi.org/10.2478/gm-2019-0001

KeywordsNeighborhood number, chromatic number, neighbor chromatic number, grid graph, torus graph

© 2019 B. Chaluvaraju et al., published by Sciendo

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

Keywords
Neighborhood number, chromatic number, neighbor chromatic number, grid graph, torus graph