Another conjecture of TxGraffiti concerning zero forcing and domination in graphs

TL;DR

This paper proves a conjecture relating zero forcing and domination numbers in connected, cubic, claw-free graphs, providing a tighter upper bound and characterizing extremal cases, advancing understanding in graph theory.

Contribution

It establishes a new upper bound for the zero forcing number in specific graphs and characterizes all graphs that attain this bound, based on a conjecture generated by AI.

Findings

Proves that Z(G) ≤ γ(G) + 2 for connected, cubic, claw-free graphs.

Provides a complete characterization of graphs achieving the bound.

Improves existing upper bounds for the zero forcing number in these graph classes.

Abstract

This paper proves a conjecture generated by the artificial intelligence conjecturing program called \emph{TxGraffiti}. More specifically, we show that if $G$ is a connected, cubic, and claw-free graph, then $Z(G) \le \gamma(G) + 2$, where $Z(G)$ and $\gamma(G)$ denote the zero forcing number and the domination number of $G$, respectively. Furthermore, we provide a complete characterization of graphs that achieve this bound. Notably, this bound improves the known upper bounds for the zero forcing number of connected, cubic, and claw-free graphs.