Zero Forcing and Vertex Independence Number on Cubic and Subcubic Graphs

TL;DR

This paper investigates the relationship between zero forcing number and vertex independence number in cubic and subcubic graphs, providing bounds and specific graph families that support a conjecture from automated conjecturing tools.

Contribution

It proves that most cubic graphs satisfy Z(G) ≤ α(G) + 2, constructs an infinite family where Z(G) = α(G) + 1, and extends bounds to subcubic graphs.

Findings

Most cubic graphs satisfy Z(G) ≤ α(G) + 2

Constructed an infinite family with Z(G) = α(G) + 1

Extended bounds to 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, $\alpha(G)$, of cubic and subcubic graphs is explored. TxGraffiti conjectures that for all connected cubic graphs $G$, that are not $K_4$, $Z(G) \leq \alpha(G) + 1$. This work uses decycling partitions of upper-embeddable graphs to show that almost all cubic graphs satisfy $Z(G) \leq \alpha(G) + 2$, provides an infinite family of cubic graphs where $Z(G) = \alpha(G) + 1$, and extends known bounds to subcubic graphs.