Bounds on the burning number

Abstract

Motivated by a graph theoretic process intended to measure the speed of the spread of contagion in a graph, Bonato et al. (Burning a Graph as a Model of Social Contagion, Lecture Notes in Computer Science 8882 (2014) 13-22) define the burning number b(G) of a graph G as the smallest integer k for which there are vertices x1,…,xk such that for every vertex u of G, there is some i∈{1,…,k} with distG(u,xi)≤k−i, and distG(xi,xj)≥j−i for every i,j∈{1,…,k}. For a connected graph G of order n, they prove that b(G)≤2n−1, and conjecture b(G)≤n. We show that b(G)≤[Formula presented]⋅[Formula presented]+[Formula presented] and b(G)≤[Formula presented]+3≈1.309n+3 for every connected graph G of order n and every 0