The Burning Number Conjecture is True for Trees without Degree-2 Vertices

Abstract

Graph burning is a discrete time process which can be used to model the spread of social contagion. One is initially given a graph of unburned vertices. At each round (time step), one vertex is burned; unburned vertices with at least one burned neighbour from the previous round also becomes burned. The burning number of a graph is the fewest number of rounds required to burn the graph. It has been conjectured that for a graph on n vertices, the burning number is at most \(\lceil \sqrt{n}\rceil \). We show that the graph burning conjecture is true for trees without degree-2 vertices.