Burnability of Double Spiders and Path Forests

TL;DR

This paper investigates the burning number conjecture for specific tree classes, proving it for double spiders and analyzing burning numbers of path forests, advancing understanding of contagion spread in networks.

Contribution

It verifies the burning number conjecture for double spiders and characterizes burning numbers of certain path forests, extending known results for trees.

Findings

Double spiders of order $m^2$ have burning number at most $m$.

Path forests with long shortest paths of order $m^2$ have burning number exactly $m$.

Results support the conjecture for broader classes of trees and forests.

Abstract

The burning number of a graph can be used to measure the spreading speed of contagion in a network. The burning number conjecture is arguably the main unresolved conjecture related to this graph parameter, which can be settled by showing that every tree of order $m^2$ has burning number at most $m$. This is known to hold for many classes of trees, including spiders - trees with exactly one vertex of degree greater than two. In fact, it has been verified that certain spiders of order slightly larger than $m^2$ also have burning numbers at most $m$, a result that has then been conjectured to be true for all trees. The first focus of this paper is to verify this slightly stronger conjecture for double spiders - trees with two vertices of degrees at least three and they are adjacent. Our other focus concerns the burning numbers of path forests, a class of graphs in which their burning numbers are naturally related to that of spiders and double spiders. Here, our main result shows that a path forest of order $m^2$ with a sufficiently long shortest path has burning number exactly $m$, the smallest possible for any path forest of the same order.