On the Total Set Chromatic Number of Graphs

Authors

Mark Anthony C. Tolentino, Ateneo de Manila UniversityFollow
Gerone Russel J. Eugenio, Ateneo de Manila University; Central Luzon State UniversityFollow
Mari-Jo P. Ruiz, Ateneo de Manila UniversityFollow

Publication Date

July 2022

Abstract

Given a vertex coloring c of a graph, the neighborhood color set of a vertex is defined to be the set of all of its neighbors’ colors. The coloring c is called a set coloring if any two adjacent vertices have different neighborhood color sets. The set chromatic number χ_s(G) of a graph G is the minimum number of colors required in a set coloring of G. In this work, we investigate a total analog of set colorings; that is, we study set colorings of the total graph of graphs. Given a graph G = (V, E), its total graph T(G) is the graph whose vertex set is V ∪ E and in which two vertices are adjacent if and only if their corresponding elements in G are adjacent or incident. First, we establish sharp bounds for the set chromatic number of the total graph of a graph. Furthermore, we study the set colorings of the total graph of different families of graphs.

Creative Commons License

This work is licensed under a Creative Commons Attribution 4.0 License.

Recommended Citation

Tolentino, Mark Anthony C.; Eugenio, Gerone Russel J.; and Ruiz, Mari-Jo P. (2022) "On the Total Set Chromatic Number of Graphs," Theory and Applications of Graphs: Vol. 9: Iss. 2, Article 5.
DOI: 10.20429/tag.2022.090205
Available at: https://digitalcommons.georgiasouthern.edu/tag/vol9/iss2/5