13 problems

Let $G$ be a connected graph on $n$ vertices, and let $b(G)$ denote the minimum number of rounds needed to burn all vertices of $G$. Burning Number Conjecture. … This conjecture pr…

Greedy optimality conjecture for comb graphs. If $n\leq m$, then

Tree form of the Burning Number Conjecture. Every tree on $n$ vertices satisfies

Let $f: \mathbb{N} \to \mathbb{N}$ be a strictly increasing function, let … for some $d>2$, and let $\mathcal G(f)=(G_n,n\geq 1)$. Write $P(\mathcal G(f))$ for the set of achievabl…

For each $n\geq4$, let $L_n$ denote the threshold parameter for $n$-path forests introduced in the paper. Linear upper-bound conjecture. … If true, this would give a good asymptoti…

Deficient path forest conjecture. Let $n\geq4$. If $T$ is a deficient $n$-path forest with $l_1\geq L_{n-1}$, then $T$ is impossibly burnable. The conjecture is motivated by comput…

The strengthened burning number conjecture. The tree $T$ is $m$-burnable.

Let $(r_1,\dots,r_k)$ be a sequence of non-negative real numbers satisfying … so that $(r_1,\dots,r_k)$ is convex. Let $T$ be a metric tree with total length…

Let $r>0$, let $T$ be a metric tree with total length $|T|\geq 2r$, and let $\mathcal{C}(T,0)$ denote the relevant space of covers of $T$ by radius-zero centers. Let $E\nu$ denote…

Burning Graph Conjecture. Every tree $T$ on $n$ vertices is $\{0,\ldots,\lceil\sqrt{n}\rceil-1\}$-burnable.

Burning Graph Conjecture. Every connected graph $G$ satisfies

Decomposed-spider conjecture. If

Let $G$ be a connected graph of order $n$, and let $b(G)$ denote its burning number. Bonato, Janssen, and Roshanbin's conjecture. … This conjecture would improve the general bound…