Large independent sets in triangle-free cubic graphs: beyond planarity

- Département d'Informatique, Université libre de Bruxelles
**ORCID iD:**0000-0001-8631-6222

- Département d'Informatique, Université libre de Bruxelles
**ORCID iD:**0000-0002-7157-6694

- Department of Applied Mathematics, Ghent University and University of Mons
**ORCID iD:**0000-0001-8984-2463

### Editorial introduction

The *independence ratio* of a graph is the ratio of the size of its largest independent set to its number of vertices. Trivially, the independence ratio of a k-colorable graph is at least \(1/k\) as each color class of a k-coloring is an independent set. However, better bounds can often be obtained for well-structured classes of graphs. In particular, Albertson, Bollobás and Tucker conjectured in 1976 that the independence ratio of every triangle-free subcubic planar graph is at least \(3/8\). The conjecture was proven by Heckman and Thomas in 2006, and the ratio is best possible as there exists a cubic triangle-free planar graph with 24 vertices and the independence number equal to 9.

The present article removes the planarity assumption. However, one needs to introduce an additional assumption since there are known to exist six 2-connected (non-planar) triangle-free subcubic graphs with the independence ratio less than \(3/8\). Bajnok and Brinkmann conjectured that every 2-connected triangle-free subcubic graph has the independence ratio at least \(3/8\) unless it is one of the six exceptional graphs. Fraughnaugh and Locke proposed a stronger conjecture: every triangle-free subcubic graph that does not contain one of the six exceptional graphs as a subgraph has independence ratio at least \(3/8\). The authors prove these two conjectures, which implies in particular the result by Heckman and Thomas.