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A graph G is said to be 1-planar if it can be drawn on the sphere or plane so that any edge of G has at most one crossing point with another edge. Moreover, G is called an optimal 1-planar graph if $$|E(G)| = 4|V(G)|-8$$ |E(G)|=4|V(G)|-8 . In this paper, we investigate the matching extendability of optimal 1-planar graphs. It is shown that every optimal 1-planar graph G of even order is 2-extendable...
We show that, for any given non-spherical orientable closed surface F2, there exists an optimal 1-planar graph which can be embedded on F2 as a triangulation. On the other hand, we prove that there does not exist any such graph for the nonorientable closed surfaces of genus at most 3.
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