Hopping forcing number in random d-regular graphs

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 ≥ 2 .