A new note on 1-planar graphs with minimum degree 7

A graph is 1-planar if it can be drawn in the plane such that each edge is crossed at most once. It is well-known that each 1-planar graph has its minimum degree at most 7. Recently, Biedl (2021) showed that any 1-planar graph with minimum degree 7 has at least 24 vertices. In this paper, we characterize 1-planar graphs with 24 vertices and minimum degree 7. Furthermore, we prove that any 1-planar graph of odd order with minimum degree 7 has at least 29 vertices, and the lower bound is tight. In addition, we also characterize 5-connected 7-regular maximal 1-plane graphs. (c) 2024 Elsevier B.V. All rights reserved.