A graph is outer-1-planar if it has a drawing in the plane so that its vertices are on the boundary face and each edge is crossed at most once. Zhang (2013) proved that the total chromatic number of every outer-1-planar graph with maximum degree △ ≥ 5 is △ + 1, and showed that there are graphs with maximum degree 3 and total chromatic number 5. For outer-1-planar graphs with maximum degree 4, Zhang (2017) confirmed that its total chromatic number is at most 5 if it admits an outer-1-planar drawing in the plane so that any two pairs of crossing edges share at most one common end vertex. In this paper, we prove that the total chromatic number of every Anicop graph with maximum degree 4 is at most 5, where an Anicop graph is an outer-1-planar graph that admits a drawing in the plane so that if there are two pairs of crossing edges sharing two common end vertices, then any of those two pairs of crossing edges would not share any end vertex with some other pair of crossing edges. This result generalizes the one of Zhang (2017) and moves a step towards the complete solving of the cold case.
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