The burning number conjecture holds for trees of order $n$ with at most $\left\lfloor \sqrt{n-1}\right\rfloor$ degree-2 vertices

Abstract

Inspired by the spread of information in social networks and graph-theoretic processes such as Firefighting and graph cleaning, Bonato, Janssen and Roshanbin introduced in 2016 the burning number $b(G)$ of any finite graph $G$. They conjectured that $b(G)\le \lceil n^\frac{1}{2}\rceil$ holds for all connected graphs $G$ of order $n$, and observed that it suffices to prove the conjecture for all trees. In 2024, Murakami confirmed the conjecture for trees without degree-2 vertices. In this paper, we prove that for all trees $T$ of order $n$ with $n_2$ degree-2 vertices, $$b(T)\le \left\lceil \left(n+n_2-\left\lceil\sqrt{n+n_2+0.25}-1.5\right\rceil\right)^{\frac{1}{2}}\right\rceil.$$ Hence, the conjecture holds for all trees of order $n$ with at most $\left\lfloor \sqrt{n-1}\right\rfloor$ degree-2 vertices.