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

TL;DR

This paper proves a conjecture that forbidding a specific small family of nonplanar subgraphs in triangle-free cubic graphs guarantees a large independent set, extending known results beyond planarity constraints.

Contribution

It confirms that excluding a family of six nonplanar subgraphs ensures large independent sets in triangle-free cubic graphs, relaxing planarity requirements.

Findings

Proved the conjecture by Fraughnaugh and Locke.

Established that excluding the six graphs guarantees large independent sets.

Confirmed the conjecture for 2-connected graphs with exceptions.

Abstract

Every $n$-vertex planar triangle-free graph with maximum degree at most $3$ has an independent set of size at least $\frac{3}{8}n$. This was first conjectured by Albertson, Bollob\'as and Tucker, and was later proved by Heckman and Thomas. Fraughnaugh and Locke conjectured that the planarity requirement could be relaxed into just forbidding a few specific nonplanar subgraphs: They described a family $\mathcal{F}$ of six nonplanar graphs (each of order at most $22$) and conjectured that every $n$-vertex triangle-free graph with maximum degree at most $3$ having no subgraph isomorphic to a member of $\mathcal{F}$ has an independent set of size at least $\frac{3}{8}n$. In this paper, we prove this conjecture. As a corollary, we obtain that every $2$-connected $n$-vertex triangle-free graph with maximum degree at most $3$ has an independent set of size at least $\frac{3}{8}n$, with the exception of the six graphs in $\mathcal{F}$. This confirms a conjecture made independently by Bajnok and Brinkmann, and by Fraughnaugh and Locke.