37 problems

For every finite connected graph $G$, the stacking and clearing thresholds are determined by almost stacked configurations:…

Let $G$ be a graph with more than one vertex. Write $\lambda(G)$ for its cover pebbling number and $\psi(G)$ for its domination cover pebbling number. Cover-to-domination cover peb…

Strong hub cover pebbling conjecture for cycles.

Strong Target Conjecture. Every graph $G$ satisfies

Let $G$ be a graph of diameter $3$ with $n$ vertices, and let $\Omega_\omega(G)$ denote the $\omega$-cover pebbling parameter used in the paper. For $\omega\geq 1$, diameter-three…

A graph is greedy if every configuration of $\pi(G)$ pebbles can be solved at any specified root using only greedy pebbling steps. It is tree-solvable if every configuration of siz…

Extremal configuration conjecture. There exists a non-$t$-solvable configuration $C$ of $k-1$ pebbles on $G$ such that, for every vertex $i$, $c_i$ is either $0$ or $p_i-1$, except…

Let $G=(V,E)$ be a graph. A distribution of pebbles on $G$ is an initial arrangement of pebbles on a subset of vertices, and its support is that subset. Let $\gamma(G)$ denote the…

Let $d\ge 1$. For a finite graph $\mathbf{G}$, let its diameter be the maximum distance between two vertices, let its connectivity be the minimum number of vertices whose removal d…

Let ${\cal G}=\{G_1,\ldots,G_n,\ldots\}$ be a graph sequence, and let $th({\cal G})$ denote its threshold when one exists. Threshold existence conjecture. Every graph sequence…

Let $t_1$ and $t_2$ be functions such that $t_1\in\Omega(n^{1/2})$, $t_2\in O(n)$, and $t_1\ll t_2$. Threshold-range conjecture. There is a graph sequence…

Let $G_1$ and $G_2$ be graphs, let $p_1,p_2$ be positive integers, and let $f_{(p_1,p_2)}$ denote the corresponding generalized pebbling number, with $f_{p_i}(G_i)$ the one-factor…

Let $T$ be a finite connected tree. Stacking–estimating conjecture. … The statement is presented as an immediate consequence of the preceding theorem under ASH, which gives the cor…

Weak Target Conjecture. Every graph $G$ satisfies

Tree-strategy optimality conjecture for $B_4$. The bound

Validity conjecture. The weight function $w_n$ is valid.

For graphs $G_1$ and $G_2$, let $P_{1/2}(G)$ denote the smallest number of pebbles in a randomly generated configuration that is solvable with probability at least $1/2$. Probabili…

Generalized weighted-star pebbling conjecture.

Let $G$ be a graph on $n$ vertices, and let $\pi^{\sf c}(G)$ denote its cop-pebbling number. Cop-pebbling bound conjecture. Every graph $G$ on $n$ vertices satisfies … This conject…

Interior-vertex lower-bound conjecture. For every interior vertex $X$,

Let $G$ be a graph and let $D$ be a target distribution on $G$. Write $|D|$ for the total demand, $\pi(G,D)$ for the smallest integer $m$ such that every configuration of $m$ pebbl…

Let $G$ and $H$ be graphs. A nonnegative function on a graph assigns a nonnegative real number to each vertex. Let $\omega_1$ be a nonnegative function on $G$ and $\omega_2$ a nonn…

Direct-product pebbling conjecture.

Strong-product pebbling conjecture. For any connected graphs $G$ and $H$,

Multiples-of-seven staircase conjecture. For all $k\ge 1$,