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arXiv · 2607.23664

A counterexample to the zero forcing versus independence conjecture for cubic and subcubic graphs

Mikko Fischer
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Abstract

We exhibit a connected graph on 24 vertices with maximum degree 3, independence number 9 and zero forcing number 11, refuting a 2017 conjecture of TxGraffiti recorded as Conjecture 2 of the survey of Davila, Brimkov and Pepper. The same construction with a different gadget gives a connected cubic graph on 36 vertices with independence number 15 and zero forcing number 17; the conjecture therefore fails also in the cubic form in which the survey's Lean 4 appendix states it. In particular Z <= alpha + 1 is not a universal bound for connected cubic graphs, and the value Z = alpha + 2 is attained.

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Mikko Fischer. 2026-07-26. A counterexample to the zero forcing versus independence conjecture for cubic and subcubic graphs. https://arxiv.org/abs/2607.23664

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