Hopping Forcing Number in Random $d$-regular Graphs

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 $d \geq 2$.

Figures (3)

• Figure 1.1: Comparison of upper and lower bounds for the hopping number for small and large values of $d$.
• Figure 6.1: Function $h_3(x)$ and $h_{10}(x)$.
• Figure 7.1: Evolution of $Y_i(t)$, $0\leq i\leq3$ for $d=3$ using the Differential Equation Method.

Theorems & Definitions (16)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Theorem 1.5
• Lemma 2.1: friedman2008proof
• Lemma 2.2: Expander Mixing Lemma
• Theorem 2.3: Differential Equation Method, Warnke2020
• Remark 2.4
• proof : Proof of Theorem \ref{['thm:contiguous_3']}