Burning Spiders

Abstract

Graph burning is a graph process modeling the spread of social contagion. Initially all the vertices of a graph G are unburned. At each step an unburned vertex is put on fire and the fire from burned vertices of the previous step spreads to their adjacent unburned vertices. This process continues till all vertices are burned. The burning number b(G) of the graph is the minimum number of steps required to burn all the vertices in the graph. The burning number conjecture by Bonato et al. states that for a connected graph G of order n, its burning number $b(G) \le \lceil \sqrt{n}~\rceil$. It is easy to observe that in order to burn a graph it is enough to burn its spanning tree. Hence it suffices to prove that for any tree T of order n, its burning number $b(T) \le \lceil \sqrt{n}~\rceil$. A spider S is a tree with one vertex of degree at least 3 and all other vertices with degree at most 2. Here we prove that for any spider S of order n, its burning number $b(S) \le \lceil \sqrt{n}~\rceil$.