Semi-total graph colourings, the beta parameter, and total chromatic number

A semi-total colouring, @m, of a graph G with maximum degree @D uses @D+1 colours and has the properties of a total colouring except that adjacent vertices need not have distinct colours. A beta edge is an edge v"1v"2 such that @m(v"1)=@m(v"2). The beta parameter of G, @b(G), is the minimum number of beta edges in any semi-total colouring of G. A critical edge of G is an edge whose deletion reduces the total chromatic number of G to @D+1. We derive the beta parameters of cycle and complete graphs; a quadratic bound for @b in terms of @D for any graph with a critical edge; and a log-linear bound for any graph with a critical edge that is not contained in any triangle.