On the Burning Number of $p$-Caterpillars

TL;DR

This paper investigates the burning number in graphs, proving the conjecture for caterpillars and certain trees, and analyzing computational complexity for specific graph classes, including caterpillars with maximum degree three.

Contribution

It provides new proofs for the burning number conjecture on caterpillars and extends results to generalized caterpillars and trees with many leaves, also analyzing complexity for degree-restricted caterpillars.

Findings

Proved the conjecture for ordinary and generalized caterpillars.

Established NP-completeness for caterpillars with maximum degree three.

Presented two different proofs for caterpillars.

Abstract

The burning number is a recently introduced graph parameter indicating the spreading speed of content in a graph through its edges. While the conjectured upper bound on the necessary numbers of time steps until all vertices are reached is proven for some specific graph classes it remains open for trees in general. We present two different proofs for ordinary caterpillars and prove the conjecture for a generalised version of caterpillars and for trees with a sufficient amount of leaves. Furthermore, determining the burning number for spider graphs, trees with maximum degree three and path-forests is known to be $\mathcal{NP}$-complete, however, we show that the complexity is already inherent in caterpillars with maximum degree three.