Burning games on strong path products

TL;DR

This work studies contagion-like spreading on graphs via burning and cooling, focusing on the strong product family $P_n^{\boxtimes d}$ to model multi-dimensional diffusion. It establishes the cooling number $\mathrm{CL}(P_n^{\boxtimes d}) = n$ and develops a tiling-based approach, coupled with Euler–Maclaurin analysis and SI-interpolators, to bound the burning number $b(P_n^{\boxtimes d})$ from below using the root $x^*$ of a rational polynomial $q(x)$ involving Bernoulli numbers; closed forms are provided for $d=2,3$, and a conjecture $b(P_n^{\boxtimes d}) = \Theta(n^{d/(d+1)})$ is proposed. The paper further introduces liminal burning with a two-player dynamic, defining the threshold $k^*$ as the smallest $k$ with $b_k(G)=b(G)$, and proves exact results for paths, notably $b_2(P_n)=\lceil (n+2)/3 \rceil$, plus bounds on $k^*$ framed by tiling feasibility. Overall, the results connect discrete diffusion on high-dimensional graph products to geometric tiling and analytic tools, yielding sharp bounds and guiding conjectures for burning in complex networks.

Abstract

Burning and cooling are diffusion processes on graphs in which burned (or cooled) vertices spread to their neighbors with a new source picked at discrete time steps. In burning, the one tries to burn the graph as fast as possible, while in cooling one wants to delay cooling as long as possible. We consider $d$-fold strong products of paths, which generalize king graphs. The propagation of these graphs is radial, and models local spread of contagion in an arbitrary number of dimensions. We reduce the problem to a geometric tiling problem to obtain a bound for the burning number of a strong product of paths by a novel use of an Euler-Maclaurin formula, which is sharp under certain number theoretic conditions. Additionally, we consider liminal burning, which is a two-player perfect knowledge game played on graphs related to the effectiveness of controlled spread of contagion throughout a network. We introduce and study the number $k^*$, the smallest $k$ such that $b_{k}(G) = b(G)$.

Figures (2)

• Figure 1: Liminal burning numbers of $G$
• Figure 2: Cooling sequences for $n=7,8$ in a 2-face of cube defined by $P_n^{\boxtimes d}$.

Theorems & Definitions (17)

• Theorem 1.1
• Theorem 1.2
• Proposition 2.1
• proof
• Lemma 2.2
• proof : Proof of Theorem \ref{['thm:burning strong']}
• Corollary 2.3
• proof
• Lemma 2.4
• proof