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 χs(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.
Tolentino, M.A.C., Eugenio, G.R.J., & Ruiz, M.J.P. (2022). On the total set chromatic number of graphs. Theory and Applications of Graphs, 9(2). https://doi.org/10.20429/tag.2022.090205