Bounds on the burning number

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)≤3219⋅n1−ϵ+2719ϵ and b(G)≤12n7+3≈1.309n+3 for every connected graph G of order n and every 0<ϵ<1. For a tree T of order n with n2 vertices of degree 2, and n≥3 vertices of degree at least 3, we show b(T)≤(n+n2)+14+12 and b(T)≤n+n≥3. Furthermore, we characterize the binary trees of depth r that have burning number r+1.