Complexity of image-total labelling Original Research Article

A image-total labelling of a graph image is a total colouring image from image into image such that image whenever an edge image is incident to a vertex image. The minimum image for which image admits a image-total labelling is denoted by image. The case image corresponds to the usual notion of total colouring, which is NP-hard to compute even for cubic bipartite graphs [C.J. McDiarmid, A. Sánchez-Arroyo, Total colouring regular bipartite graphs is NP-hard, Discrete Math. 124 (1994), 155–162]. In this paper we assume image. It is easy to show that image, where image is the maximum degree of image. Moreover, when image is bipartite, image is an upper bound for image, leaving only two possible values. In this paper, we completely settle the computational complexity of deciding whether image is equal to image or to image when image is bipartite. This is trivial when image, polynomial when image and image, and NP-complete in the remaining cases.