Burning Number Conjecture (Upper Bound by sqrt(n))

Establish that for every connected graph G with n vertices, the burning number b(G)—defined as the minimum integer k for which there exists a sequence of vertices (u1, …, uk) such that every vertex v is within graph distance at most k−i from some ui—satisfies b(G) ≤ ⌈√n⌉.

Background

The graph burning problem (GBP) seeks the shortest sequence of vertices that, under a discrete-time contagion process, burns all vertices of a graph; the minimum sequence length is the burning number b(G). The burning number conjecture (BNC) asserts a universal upper bound of ⌈√n⌉ for connected graphs with n vertices and is a central question in GBP.

Despite progress, the best-known general upper bound is currently ⌈(4n/3)^{1/2}⌉ + 1. The paper studies a greedy heuristic for GBP and its relation to clustered maximum coverage, but the conjecture itself remains open and is independent of the heuristic's contributions.