A note on 1-planar graphs with minimum degree 7

TL;DR

This paper proves that 1-planar graphs with minimum degree 7 must have at least 24 vertices, establishing a tight bound and deepening understanding of their structural properties.

Contribution

It establishes the minimum number of vertices in 1-planar graphs with minimum degree 7, providing a tight bound that was previously unknown.

Findings

Any 1-planar graph with minimum degree 7 has at least 24 vertices.

The bound of 24 vertices is tight, meaning it cannot be improved.

This result refines the structural understanding of 1-planar graphs.

Abstract

It is well-known that 1-planar graphs have minimum degree at most 7, and not hard to see that some 1-planar graphs have minimum degree exactly 7. In this note we show that any such 1-planar graph has at least 24 vertices, and this is tight.