Kostochka’s bipartite cubic 1/3-Conjecture

Establish that every cubic bipartite graph G of order n satisfies the domination bound γ(G) ≤ n/3.

Background

Prompted by a question posed by Kostochka (2009), this problem was later formalized as a conjecture (He16). It asks whether bipartiteness alone forces the optimal one-third domination bound in cubic graphs, without a girth condition.

The authors prove the conjecture under added cycle exclusions (no 4- and no 8-cycles), but the general case remains open and of central interest in domination theory for regular bipartite graphs.