A criterion for the planarity of the total graph of a graph

This is the authors' abstract. We don't add key points for this paper.

Two well-known numbers associated with a graph G (finite and undirected with no loops or multiple lines) are the (point) chromatic and the line chromatic number of G (see (2)). With G there is associated a graph L(G), called the line-graph of G, such that the line chromatic number of G is the same as the chromatic number of L(G). This concept was originated by Whitney (9) in 1932. In 1963, Sedlâček (8) characterized graphs with planar line-graphs. In this note we introduce the notions of the total chromatic number and the total graph of a graph, and characterize graphs with planar total graphs.