Total b-chromatic Colouring of Graphs

A b-chromatic colouring of a graph $G$ is a proper $k$-colouring of the vertices of $G$, for some integer $k$, such that, for each colour $i$ ($1\leq i\leq k$), there exists a vertex $v$ of colour $i$ such that $v$ is adjacent to a vertex of colour $j$, for each $j$ ($1\leq j\leq k$, $j\neq i$). The b-chromatic number of $G$ is the maximum integer $k$ such that $G$ admits a b-chromatic colouring using $k$ colours. In this paper we introduce the concept of a total b-chromatic colouring, which extends the notion of b-chromatic colourings to both vertices and edges in a graph. We show that the problem of computing the total b-chromatic number is NP-hard in general graphs. On the other hand for a subclass of caterpillars we give a polynomial-time algorithm to compute the total b-chromatic number, and indeed a total b-chromatic colouring with the maximum number of colours.