Incidence Coloring of Outer-1-planar Graphs

A graph is outer-1-planar if it can be drawn in the plane so that all vertices lie on the outer-face and each edge crosses at most one another edge. It is known that every outer-1-planar graph is a planar partial 3-tree. In this paper, we conjecture that every planar graph G has a proper incidence (Δ(G) + 2)-coloring and confirm it for outer-1-planar graphs with maximum degree at least 8 or with girth at least 4. Specifically, we prove that every outer-1-planar graph G has an incidence (Δ(G) + 3, 2)-coloring, and every outer-1-planar graph G with maximum degree at least 8 or with girth at least 4 has an incidence (Δ(G) + 2, 2)-coloring.