An upper bound for total colouring of graphs (Q686500)

Given a simple graph \(G=(V,E)\), a total coloring of \(G\) is an assignment of colors to the elements of \(V \cup E\) in such a way that no two adjacent or incident elements receive the same color. The total chromatic number \(\chi_ T(G)\) is the least number of colors in a total coloring of \(G\). There is a thirty years old conjecture that \(\chi_ T(G) \leq\Delta+2\), where \(\Delta\) is the maximum degree of a vertex in \(G\). In the paper the authors give a new upper bound on the total chromatic number showing that \(\chi_ T(G)<7\Delta/5+3\).