Bounds on the burning numbers of spiders and path-forests

TL;DR

This paper proves the conjectured upper bound on the burning number for spider graphs and introduces new bounds for path-forests, along with an approximation algorithm for their burning number.

Contribution

It establishes the conjectured bound for spider graphs and develops new bounds and an approximation algorithm for path-forests.

Findings

Proved the burning number bound for spider graphs.

Developed new bounds for path-forests.

Created a 3/2-approximation algorithm for path-forest burning numbers.

Abstract

Graph burning is one model for the spread of memes and contagion in social networks. The corresponding graph parameter is the burning number of a graph $G$, written $b(G)$, which measures the speed of the social contagion. While it is conjectured that the burning number of a connected graph of order $n$ is at most $\lceil \sqrt{n} \rceil$, this remains open in general and in many graph families. We prove the conjectured bound for spider graphs, which are trees with exactly one vertex of degree at least 3. To prove our result for spiders, we develop new bounds on the burning number for path-forests, which in turn leads to a $\frac 3 2$-approximation algorithm for computing the burning number of path-forests.