Zero Forcing Number of Random Regular Graphs

TL;DR

This paper investigates the zero forcing number in random regular graphs, providing improved bounds and introducing a degree-greedy algorithm analyzed via differential equations to find small zero forcing sets.

Contribution

It offers new bounds for the zero forcing number in random regular graphs and proposes a novel degree-greedy algorithm with theoretical analysis.

Findings

Improved bounds on zero forcing number for random regular graphs

Development of a degree-greedy algorithm for zero forcing sets

Analysis of the algorithm using differential equations

Abstract

The zero forcing process is an iterative graph colouring process in which at each time step a coloured vertex with a single uncoloured neighbour can force this neighbour to become coloured. A zero forcing set of a graph is an initial set of coloured vertices that can eventually force the entire graph to be coloured. The zero forcing number is the size of the smallest zero forcing set. We explore the zero forcing number for random regular graphs, improving on bounds given by Kalinowski, Kam\u{c}ev and Sudakov. We also propose and analyze a degree-greedy algorithm for finding small zero forcing sets using the differential equations method.