41 problems

Let $G$ be a connected graph with $n$ vertices. Burning number conjecture. The graph $G$ can be burned in at most … steps. This conjecture concerns a fundamental upper bound for th…

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…

Let $G$ be a connected graph of order $n$. The burning number $b(G)$ is the minimum length of a burning sequence for $G$, where a burning sequence is a sequence of sources that pro…

Burning Number Conjecture.

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

Burning number conjecture.

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

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…

NP-completeness conjecture for $(k,d)$-special graphs. Deciding whether a graph is $(k,d)$-special is NP-complete in general.

Asymptotic hypercube burning conjecture. If $k$ is a positive integer, then there exists an integer $n_k$ such that

A $p$-caterpillar is a caterpillar in which the vertices not on the spine have distance at most $p$ from the spine. Let the shift and jump operations be the operations developed in…

Let $G$ be a connected graph on $n$ vertices, and let $b_{\rm g}(G)$ denote its Burner-start game burning number. Let $P_n$ be the path on $n$ vertices. The burning-game path extre…

Let $H$ be a hypergraph. For each proportion $p\in(0,1)$, let $b_p(H)$ and $b_{L,p}(H)$ denote the burning and lazy burning numbers, respectively; their constant-value subintervals…

Let $H$ be a connected hypergraph, and let $pin(0,1)\cap\mathbb{Q}$. Assume that $H$ contains no non-flammable edges. Proportion-based lazy burning conjecture. Then … The bound is…

Let $G$ be a finite graph with $n$ vertices, and let $\beta(G)$ denote its burning number: the minimum length of a sequence $x_0,x_1,\ldots,x_b$ such that … where $\Gamma_k(x)$ is…

Let $G=(V,E)$ be an arbitrary undirected graph with $p\geq 1$ connected components, and let $n_1,n_2,\dots,n_p$ be the numbers of vertices in these components. Denote by $b(G)$ the…

Moghbel's total burning conjecture. The total burning number satisfies

Burning number conjecture. The bound can be improved to

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…

Let $G$ be a connected graph of order $N$, and let $b(G)$ denote its burning number. Burning number conjecture. The burning number of every connected graph of order $N$ is at most…

Let $G$ be a connected graph on $n$ vertices. Its burning number $b(G)$ is the smallest positive integer $k$ such that the vertex set of $G$ can be covered by balls of radii…

For a graph $G$, its burning number $b(G)$ is the minimum number of rounds required to burn all vertices of $G$. A path is a graph whose vertices can be ordered so that consecutive…

Let $G$ be an undirected finite graph without loops or multiple edges, and let $b(G)$ denote its burning number. If $G$ is connected and has order $m$, then the burning number conj…

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…