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An incidence of a graph $G$ is a pair $(v,e)$ where $v$ is a vertex of~$G$ and $e$ is an edge of~$G$ incident with $v$. Two incidences $(v,e)$ and $(w,f)$ of~$G$ are adjacent whenever (i) $v=w$, or (ii) $ e=f$, or (iii) $vw=e$ or $f$. An incidence $p$-colouring of~$G$ is a mapping from the set of incidences of~$G$ to the set of colours $\{1,\dots,p\}$ such that every two adjacent incidences receive distinct colours. Incidence colouring has been introduced by Brualdi and Quinn Massey in 1993 and, since then, studied by several authors.
In this paper, we introduce and study the strong version of incidence colouring, where incidences adjacent to a same incidence must also get distinct colours. We determine the exact value of -- or upper bounds on -- the strong incidence chromatic number of several classes of graphs, namely cycles, wheel graphs, trees, ladder graphs and subclasses of Halin graphs.
