Dice Question Streamline Icon: https://streamlinehq.com

Zero forcing versus independence in subcubic graphs (excluding K4)

Prove that for every connected finite simple undirected graph G with maximum degree Δ(G) ≤ 3 and G not isomorphic to K4, the zero forcing number Z(G) satisfies Z(G) ≤ α(G) + 1; moreover, demonstrate that this bound is sharp.

Information Square Streamline Icon: https://streamlinehq.com

Background

Zero forcing is a propagation process defining the zero forcing number Z(G), with applications to minimum rank and control theory. The conjecture relates Z(G) to the independence number α(G) for connected subcubic graphs, excluding K4.

The inequality has been verified or strengthened under additional structural restrictions (e.g., claw-free graphs) and is known to be sharp on infinite structured families, but remains unproved in full generality for all connected subcubic graphs not isomorphic to K4.

References

The following conjecture was generated using a linear programming-based approach applied to cubic graphs. If $G \not\cong K_4$ is a connected graph with $\Delta(G) \leq 3$, then $Z(G) \le \alpha(G) + 1$, and this bound is sharp.

— In Reverie Together: Ten Years of Mathematical Discovery with a Machine Collaborator (2507.17780 - Davila et al., 23 Jul 2025) in Section "Open conjectures of TxGraffiti (2016–2025)", Subsection "The (α, Z)-conjecture", Conjecture [TxGraffiti -- Open Since 2017]