Total colorings of some join graphs (Q2747198)

The authors consider simultaneous colourings of the vertices and edges of a finite simple graph \(G\) so that no two edges incident with the same vertex receive the same colour, no two adjacent vertices receive the same colour, and no incident edge and vertex receive the same colour. \(\chi_T(G)\) means the least number of colours needed for such type of colouring of \(G\). A graph \(G\) is called Type 1 if \(\chi_T(G)= \Delta(G)+ 1\) and Type 2 if \(\chi_T(G)= \Delta(G)+2\), where \(\Delta(G)\) is the maximum vertex degree of \(G\). The authors classify the join graphs of the form \(G_1+ G_2\), where \(G_1\) and \(G_2\) are graphs with maximum degree at most 2, according to Type 1 (Type 2).