Conjectured square-root upper bound for the burning number of arbitrary graphs

Establish that for any undirected graph G with p connected components whose component sizes (number of vertices) are n1, n2, ..., np, the burning number b(G) satisfies b(G) ≤ sum over i = 1 to p of ceil(sqrt(n_i)).

Background

The Graph Burning Problem (GBP) seeks a minimum-length sequence of vertices whose expanding neighborhoods cover all vertices of the graph; the minimum length is called the burning number b(G). For an arbitrary undirected graph G with p connected components of sizes n1, ..., np, the best known general upper bound is b(G) ≤ p + sum_{i=1}^p ceil((4 n_i / 3)^{1/2}).

A longstanding conjecture proposes a tighter bound b(G) ≤ sum_{i=1}^p ceil(sqrt(n_i)). This conjecture has remained unresolved since the GBP was introduced and, if proven, would improve the general understanding of contagion spread rates by tightening the universal upper bound on b(G).