Verstraete’s 1/3-Conjecture for cubic graphs of girth at least 6

Prove that for every cubic graph G on n vertices with girth g(G) ≥ 6, the domination number satisfies γ(G) ≤ n/3.

Background

Löwenstein and Rautenbach (2008) showed that γ(G) ≤ n/3 for cubic graphs with girth at least 83, initiating efforts to lower the girth threshold. Verstraete (2010) conjectured that the threshold could be reduced dramatically to girth 6.

This conjecture is consistent with known counterexamples to Reed’s earlier conjecture (which are confined to small girth), and has motivated substantial work; the present paper confirms the conjecture under additional exclusions of 7- and 8-cycles.