Zero Forcing in Claw-Free Cubic Graphs

TL;DR

This paper establishes a new upper bound for the zero forcing number in connected, cubic, claw-free graphs, linking it to the independence number and matching number, which improves previous bounds and enhances understanding of these graph invariants.

Contribution

It proves that for such graphs, the zero forcing number is at most the independence number plus one, and also bounded by the matching number, providing tighter bounds than previously known.

Findings

Z(G)  lpha(G) + 1 for connected, cubic, claw-free graphs not equal to K4.

Z(G)  rac{2}{5}n + 1 for graphs of order n.

Z(G)  lpha'(G), the matching number.

Abstract

The zero forcing number of a simple graph, written $Z(G)$, is a NP-hard graph invariant which is the result of the zero forcing color change rule. This graph invariant has been heavily studied by linear algebraists, physicists, and graph theorist. It's broad applicability and interesting combinatorial properties have attracted the attention of many researchers. Of particular interest, is that of bounding the zero forcing number from above. In this paper we show a surprising relation between the zero forcing number of a graph and the independence number of a graph, denoted $\alpha(G)$. Our main theorem states that if $G \ne K_4$ is a connected, cubic, claw-free graph, then $Z(G) \le \alpha(G) + 1$. This improves on best known upper bounds for $Z(G)$, as well as known lower bounds on $\alpha(G)$. As a consequence of this result, if $G \ne K_4$ is a connected, cubic, claw-free graph with order $n$, then $Z(G) \le \frac{2}{5}n + 1$. Additionally, under the hypothesis of our main theorem, we further show $Z(G) \le \alpha'(G)$, where $\alpha'(G)$ denotes the matching number of $G$.