Burning rooted graph products

TL;DR

This paper proves the Burning Number Conjecture for comb graphs, a class of rooted graph products, and provides exact formulas and asymptotic behavior for their burning numbers using a constructive greedy algorithm.

Contribution

It establishes the BNC for comb graphs and derives precise formulas and asymptotics, extending understanding of influence spread in hierarchical network models.

Findings

BNC holds for all comb graphs.

Exact formulas for burning numbers in various regimes.

Asymptotic order of burning numbers determined.

Abstract

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.