Smallest Cubic Non-1-Planar Graphs

2026-09-22

AI Summary

研究解决了寻找最小的非1-平面三次图的问题，通过计算机辅助和引入k-灵活性概念的方法，证明了最小的非1-平面三次图有30个顶点。

Abstract

A graph is 1-planar if it has a drawing in which every edge is crossed at most once. We show that the smallest cubic non-1-planar graphs have $30$ vertices. Two such graphs are the Tutte-Coxeter graph of girth eight and a graph of girth seven that we call the Byte graph. Every subcubic graph with fewer than $30$ vertices is 1-planar. Our proof is computer-assisted, but directly testing all relevant graphs is impractical. To establish non-1-planarity of the two graphs, we extend a SAT-based solver with a custom clause propagator based on separating cycles and a case split based on graph automorphisms, allowing independent cases to be solved in parallel. To show that all smaller subcubic graphs are 1-planar, we introduce the concept of $k$-flexibility: every set of at most $k$ prescribed edges can remain uncrossed in some 1-planar drawing. We use this property to reconstruct 1-planar drawings of larger graphs from drawings of smaller $k$-flexible graphs. This replaces exhaustive testing of more than forty billion cubic graphs with computations on far fewer graphs of smaller order.

Problem

• 1-planar
• cubic graph
• Tutte-Coxeter graph
• Byte graph

Innovation

• k-flexibility
• SAT-based solver
• clause propagator
• graph automorphisms