Zero Forcing and Vertex Independence Number on Cubic and Subcubic Graphs

Abstract

Motivated by a conjecture from the automated conjecturing program TxGraffiti, in this paper the relationship between the zero forcing number, $Z(G)$, and the vertex independence number, $\hat{I}\pm (G)$, of cubic and subcubic graphs is explored. TxGraffiti conjectures that for all connected cubic graphs $G$, that are not $K_4$, $Z(G)\le \hat{I}\pm (G)+1$. This work uses decycling partitions of upper-embeddable graphs to show that almost all cubic graphs satisfy $Z(G)\le \hat{I}\pm (G)+2$, provides an infinite family of cubic graphs where $Z(G)=\hat{I}\pm (G)+1$, and extends known bounds to subcubic graphs.