Graph Burning: Tight Bounds on the Burning Numbers of Path Forests and Spiders

TL;DR

This paper establishes tight bounds on the burning numbers of path forests and spiders, advancing understanding of graph burning processes and their relation to graph structure, with implications for social contagion modeling.

Contribution

The paper provides the first tight bounds for the burning numbers of spiders and path forests, considering their structural properties and improving previous bounds related to the conjecture.

Findings

Tight upper bounds for burning numbers of spiders based on the number of arms.

Tight bounds for the order of path forests to be burned within a given number of rounds.

Strengthened the connection between graph structure and burning number bounds.

Abstract

In 2016, Bonato, Janssen, and Roshanbin introduced graph burning as a discrete process that models the spread of social contagion. Although the burning process is a simple algorithm, the problem of determining the least number of rounds needed to completely burn a graph, called the burning number of the graph, is NP-complete even for elementary graph structures like spiders. An early conjecture that every connected graph of order square of m can be burned in at most m rounds is the main motivator of this study. Attempts to prove the conjecture have resulted in various upper bounds for the burning number and validation of the conjecture for certain elementary classes of graphs. In this work, we find a tight upper bound for the order of a spider for it to be burned within a given number of rounds. Our result shows that the tight bound depends on the structure of the spider under consideration, namely the number of arms. This strengthens the previously known results on spiders in relation to the conjecture. More importantly, this opens up potential enquiry into the connection between burning numbers and certain characteristics of graphs. Finally, a tight upper bound for the order of a path forest for it to be burned within a given number of rounds is obtained, thus completing previously known partial corresponding results.