Conflict-free incidence coloring of outer-1-planar graphs

An incidence of a graph $G$ is a vertex-edge pair $(v,e)$ such that $v$ is incidence with $e$. A conflict-free incidence coloring of a graph is a coloring of the incidences in such a way that two incidences $(u,e)$ and $(v,f)$ get distinct colors if and only if they conflict each other, i.e.,(i) $u=v$, (ii) $uv$ is $e$ or $f$, or (iii) there is a vertex $w$ such that $uw=e$ and $vw=f$. The minimum number of colors used among all conflict-free incidence colorings of a graph is the conflict-free incidence chromatic number. A graph is outer-1-planar if it can be drawn in the plane so that vertices are on the outer-boundary and each edge is crossed at most once. In this paper, we show that the conflict-free incidence chromatic number of an outer-1-planar graph with maximum degree $\Delta$ is either $2\Delta$ or $2\Delta+1$ unless the graph is a cycle on three vertices, and moreover, all outer-1-planar graphs with conflict-free incidence chromatic number $2\Delta$ or $2\Delta+1$ are completely characterized. An efficient algorithm for constructing an optimal conflict-free incidence coloring of a connected outer-1-planar graph is given.