Burning rooted graph products

The burning number $b(G)$ of a graph $G$ is the minimum number of rounds required to burn all vertices when, at each discrete step, existing fires spread to neighboring vertices and one new fire may be ignited at an unburned vertex. This parameter measures the speed of influence propagation in a network and has been studied as a model for information diffusion and resource allocation in distributed systems. A central open problem, the Burning Number Conjecture (BNC), asserts that every graph on $n$ vertices can be burned in at most $\lceil \sqrt n\rceil$ rounds, a bound known to be sharp for paths and verified for several structured families of trees. We investigate rooted graph products, focusing on comb graphs obtained by attaching a path (a tooth'') to each vertex of a path (the spine’’). Unlike classical symmetric graph products, rooted products introduce hierarchical bottlenecks: communication between local subnetworks must pass through designated root vertices, providing a natural model for hub-and-spoke or chain-of-command architectures. We prove that the BNC holds for all comb graphs and determine the precise asymptotic order of their burning number in every parameter regime, including exact formulas in the spine-dominant case that generalize the known formula for paths. Our approach is constructive, based on an explicit greedy algorithm that is optimal or near-optimal depending on the regime.

💡 Research Summary

The paper studies the “burning number” b(G) of a graph G, a parameter that models the spread of influence (or fire) in discrete time: at each round every already‑burned vertex ignites all its neighbors, and one additional unburned vertex may be ignited. The minimum number of rounds needed to burn the whole graph is b(G). The central open problem, the Burning Number Conjecture (BNC), asserts that for any connected n‑vertex graph G we have b(G) ≤ ⌈√n⌉; this bound is tight for paths and has been proved for several families of trees and other structured graphs.

The authors introduce the rooted graph product G ∘ H. Given a graph H with a distinguished root r, the product is formed by taking one copy of G and, for each vertex v ∈ V(G), attaching a copy of H by identifying v with the root r of that copy. Consequently, every copy of H communicates with the rest of the graph only through its root, creating a hierarchical bottleneck. This contrasts with the strong (⊠) and Cartesian (□) products, where propagation can travel in many independent directions.

Theorem 1 establishes a simple bound: \