There are no Cubic Graphs on 26 Vertices with Crossing Number 10 or 11

TL;DR

This paper proves that no cubic graphs with 26 vertices have crossing number 10 or 11, confirming that the smallest such graphs with crossing number 11 have at least 28 vertices, using heuristic and elimination methods.

Contribution

It establishes the non-existence of certain cubic graphs with specific crossing numbers at 26 vertices and provides new minimal examples at larger sizes.

Findings

No cubic graphs on 26 vertices have crossing number 10 or 11.

Identified a cubic graph on 28 vertices with crossing number 10.

Discovered a cubic graph on 30 vertices with crossing number 12.

Abstract

We show that no cubic graphs of order 26 have crossing number larger than 9, which proves a conjecture of Ed Pegg Jr and Geoffrey Exoo that the smallest cubic graphs with crossing number 11 have 28 vertices. This result is achieved by first eliminating all girth 3 graphs from consideration, and then using the recently developed QuickCross heuristic to find good embeddings of each remaining graph. In the cases where the embedding found has 10 or more crossings, the heuristic is re-run with a different settings of parameters until an embedding with fewer than 10 crossings is found. We provide a minimal example of a cubic graph on 28 vertices with crossing number 10, and also exhibit for the first time a cubic graph on 30 vertices with crossing number 12, which we conjecture is minimal.