Leaky forcing and resilience of Cartesian products of $K_n$

TL;DR

This paper investigates leaky forcing in Cartesian products of complete graphs with paths and cycles, establishing 1-resilience for certain products and conjecturing higher resilience limitations.

Contribution

It proves that direct products of $K_n$ with $P_t$ and $K_n$ with $C_t$ are 1-resilient and introduces conjectures about their higher resilience.

Findings

Proves $K_n imes P_t$ and $K_n imes C_t$ are 1-resilient.

Conjectures $K_n imes P_t$ is not 2-resilient.

Explores leaky forcing and resilience in graph products.

Abstract

Zero forcing is a process on a graph $G = (V,E)$ in which a set of initially colored vertices,$B_0(G) \subset V(G)$, can color their neighbors according to the color change rule. The color change rule states that if a vertex $v$ can color a neighbor $u$ if $u$ is the only uncolored neighbor of $v$. If a vertex $v$ colors its neighbor, $u$, $v$ is said to force $u$. Leaky forcing is a recently introduced variant of zero forcing in which some vertices cannot force their neighbors, even if they satisfy the color change rule. This variation has been studied for limited families of graphs with particular structure, such as products of paths and discrete hypercubes. A concept closely related to $\ell$-leaky forcing is $\ell$-resilience. A graph is said to be $\ell$-resilient if its $\ell$-leaky forcing number equals its zero forcing number. In this paper, we prove direct products of $K_n$ with $P_t$ and $K_n$ with $C_t$ is 1-resilient and conjecture the former is not 2-resilient.