Graph Burning On Large $p$-Caterpillars

TL;DR

This paper proves the burning number conjecture for large $p$-caterpillars, a class of tree-like graphs, advancing understanding of information spread models in graph theory.

Contribution

It establishes the conjecture for a new class of graphs, large $p$-caterpillars, extending previous partial results.

Findings

Burning number conjecture holds for sufficiently large $p$-caterpillars.

Provides new bounds and techniques for analyzing graph burning.

Enhances understanding of contagion spread in complex tree structures.

Abstract

Graph burning models the spread of information or contagion in a graph. At each time step, two events occur: neighbours of already burned vertices become burned, and a new vertex is chosen to be burned. The big conjecture is known as the {\it burning number conjecture}: for any connected graph on $n$ vertices, all $n$ vertices can be burned after at most $\lceil \sqrt{n}\ \rceil$ time steps. It is well-known that to prove the conjecture, it suffices to prove it for trees. We prove the conjecture for sufficiently large $p$-caterpillars.