Hopping Forcing Number in Random $d$-regular Graphs

TL;DR

This paper investigates the hopping forcing number in random $d$-regular graphs, providing asymptotic bounds and insights into the minimum initial blue vertices needed to color the entire graph blue.

Contribution

It introduces the hopping forcing game on random $d$-regular graphs and derives asymptotic bounds for the hopping forcing number across different degrees.

Findings

Derived asymptotic upper bounds for the hopping forcing number.

Established asymptotic lower bounds for the hopping forcing number.

Provided insights into how the degree $d$ influences the forcing number.

Abstract

Hopping forcing is a single player combinatorial game in which the player is presented a graph on $n$ vertices, some of which are initially blue with the remaining vertices being white. In each round $t$, a blue vertex $v$ with all neighbours blue may hop and colour a white vertex blue in the second neighbourhood, provided that $v$ has not performed a hop in the previous $t-1$ rounds. The objective of the game is to eventually colour every vertex blue by repeatedly applying the hopping forcing rule. Subsequently, for a given graph $G$, the hopping forcing number is the minimum number of initial blue vertices that are required to achieve the objective. In this paper, we study the hopping forcing number for random $d$-regular graphs. Specifically, we aim to derive asymptotic upper and lower bounds for the hopping forcing number for various values of $d \geq 2$.