On the Total Set Chromatic Number of Graphs
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 minim...
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| Main Authors: | , , |
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| Format: | text |
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Archīum Ateneo
2022
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| Online Access: | https://archium.ateneo.edu/mathematics-faculty-pubs/216 https://archium.ateneo.edu/cgi/viewcontent.cgi?article=1217&context=mathematics-faculty-pubs |
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| Institution: | Ateneo De Manila University |
| Summary: | 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. |
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