On the Total Set Chromatic Number of Graphs

• Main Authors: Tolentino, Mark Anthony C, Eugenio, Gerone Russel J, Ruiz, Mari-Jo P
• Format: text
• Published: Archīum Ateneo 2022
• Subjects: set coloring; total graph; Mathematics; Physical Sciences and Mathematics
• Online Access: https://archium.ateneo.edu/mathematics-faculty-pubs/216; https://archium.ateneo.edu/cgi/viewcontent.cgi?article=1217&context=mathematics-faculty-pubs
• Institution: Ateneo De Manila University

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.