Bounds on the burning number

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 x(1),..,x(k) such that for every vertex u of G, there is some i is an element of {1,..., k} with distG(u, x(i)) <= k-i, and dist(G)(x(i), x(i)) >= j-i for every i,j is an element of {1,..., k}. For a connected graph G of order n, they prove that b(G) <= 2 [root n] - 1, and conjecture b(G) <= [root n]. We show that b(G) <= root 32/19.n/1-is an element of + root 27/19 is an element of and b(G) <=root 12n/7 +3 approximate to 1.309 root n+ 3 for every connected graph G of order n and every 0 < is an element of < 1. For a tree T of order n with n(2) vertices of degree 2, and n >= 3 vertices of degree at least 3, we show b(T) <=[root (n + n(2)) + 1/4 + 1/2] and b(T) <= [root n] + n(>= 3). Furthermore, we characterize the binary trees of depth r that have burning number r + 1. (C) 2017 Elsevier B.V. All rights reserved.