A graph is 1‐planar if it can be drawn in the plane such that each edge is crossed at most once. A graph, together with a 1‐planar drawing is called 1‐plane. A graph is maximal 1‐planar (1‐plane), if we cannot add any missing edge so that the resulting graph is still 1‐planar (1‐plane). Brandenburg et al. showed that there are maximal 1‐planar graphs with only 4517n+O(1)≈2.647n edges and maximal 1‐plane graphs with only 73n+O(1)≈2.33n edges. On the other hand, they showed that a maximal 1‐planar graph has at least 2813n−O(1)≈2.15n−O(1) edges, and a maximal 1‐plane graph has at least 2.1n−O(1) edges. We improve both lower bounds to 20n9≈2.22n.
Journal of Graph Theory – Wiley
Published: Jan 1, 2018
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