Graphs with burning number three

Abstract

Recently, Bonato et al. proposed the concept of the burning number of a graph to measure the speed of contagion spread on a network. They pointed out that the burning number of graph G is $b\left(G\right)\ge 2$ and showed that the graph G with burning number 2 if and only if G has maximum degree $\left|V\right(G\left)\right|-1$ or $\left|V\right(G\left)\right|-2$. Consider the problem of finding the maximum degree of a graph is solvable in polynomial time, graphs with burning number 2 can be recognized in polynomial time and thus the lower bound of burning number be improved to 3. In this paper, we characterize graphs with burning number 3 in terms of maximum degree and diameter.