On the Edge-Connectivity and Restricted Edge-Connectivity of Optimal 1-Planar Graphs

graph is called 1-planar if it can be drawn on the plane (or on the sphere) such that each edge is crossed at most once. It is known that a 1-planar graph G with at least three vertices has at most 4|V(G)|- 8 edges, so G is optimal if G has exactly 4|V(G)| - 8 edges. The vertex-connectivity of an optimal 1-planar graph has been proven to be either 4 or 6. According to the Whitney inequality, the edge-connectivity of an optimal 1-planar graph is at least 4. In this paper, we provide a more precise result. We prove that the edge-connectivity of every optimal 1-planar graph is 6. Additionally, we prove that any minimum edge-cut of an optimal 1-planar graph is a set of all the edges incident with a vertex of degree 6. In addition, we prove that an optimal 1-planar graph has the restricted edge-connectivity 8, 10 or 12.