Biplanar graphs with independence number two are 9-colorable

2026-09-23

AI Summary

研究解决了双平面图的着色问题，证明了独立数不超过2的双平面图是9-可着色的，使用SAT求解器和Lean 4验证计算结果。

Abstract

A graph is biplanar if it is the union of two planar graphs on the same vertex set. The largest chromatic number of a biplanar graph is known to lie between 9 and 12. The lower bound comes from Sulanke's graph, which has independence number 2, and a biplanar graph on 19 vertices with independence number 2 would have chromatic number at least 10. Gethner and Sulanke asked in 2009 whether such a graph exists. We show that it does not, and more generally that every biplanar graph with independence number at most 2 is 9-colorable. The proof embeds a hypothetical counterexample in the union of two sphere triangulations, enumerates with SAT modulo symmetries the 3271 graphs that pass a necessary filter for the complement of such a union, and shows with a SAT solver that none of them is such a complement; a matching argument reduces the general statement to this computation and one further case on 18 vertices. The computational part of the proof, including the completeness of the enumeration and every refutation, is checked in Lean 4, assuming three classical facts about planar graphs. The Lean development, the SAT instances, and the enumeration certificates are available on Zenodo.

Problem

• biplanar graphs
• independence number
• chromatic number

Innovation

• biplanar graphs
• 9-colorable
• independence number two
• SAT solver
• formal verification