Abstract
A {\it $(p,1)$-total labelling} of a graph $G=(V,E)$ is a total coloring $L$ from $V\cup E$ into $\{0,\dots ,l\}$ such that $|L(v)-L(e)|\geq p$ whenever an edge $e$ is incident to a vertex $v$. The minimum $l$ for which $G$ admits a $(p,1)$-total labelling is denoted by $\lambda_p(G)$. The case $p=1$ corresponds to the usual notion of total colouring, which is NP-hard to calculate even for cubic bipartite graphs~\cite{MDSA94}. We assume $p\geq 2$ in this paper. It is easy to show that $\lambda_p(G)\geq \Delta +p-1$, where $\Delta$ is the maximum degree of $G$. Moreover, when $G$ is bipartite, $\Delta +p$ is an upper bound for $\lambda_p(G)$, leaving only two possible values. In this paper, we completely settle the computational complexity of deciding whether $\lambda_p(G)$ is equal to $\Delta +p-1$ or to $\Delta +p$ when $G$ is bipartite. This is trivial when $\Delta \leq p$, polynomial when $\Delta =3$ and $p=2$, and NP-complete in the remaining cases.