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

A semi-total colouring, image, of a graph G with maximum degree image uses image colours and has the properties of a total colouring except that adjacent vertices need not have distinct colours. A beta edge is an edge image such that image. The beta parameter of G, image, 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 image. We derive the beta parameters of cycle and complete graphs; a quadratic bound for image in terms of image 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