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 Δ is either 2Δ or 2Δ + 1 unless the graph is a cycle on three vertices, and moreover, all outer-1-planar graphs with conflict-free incidence chromatic number 2Δ or 2Δ + 1 are completely characterized. An efficient algorithm for constructing an optimal conflict-free incidence coloring of a connected outer-1-planar graph is given.