59 problems

Let $G$ and $G'$ be graphs, and let $\mathcal{S}$ and $\mathcal{S}'$ be sets of distributions on $G$ and $G'$, respectively. The Cartesian product $G\Box G'$ is the graph product d…

Let $N=\{n_1,n_2,\ldots,n_q\}$ be a set of $q$ natural numbers. There is a nonempty index set $I\subset\{1,\ldots,q\}$ such that $q\mid\sum_{i\in I}n_i$. Erdős–Lemke conjecture. Th…

Pebbling-threshold spectrum conjecture. There is a graph sequence $\mathcal G$ such that

Strong Target Conjecture. Every graph $G$ satisfies

Weak 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…

For fixed $l$, let $\mathcal P_l^n$ denote the sequence of $n$-fold Cartesian powers of the path $P_l$. Let $\tau$ denote the pebbling threshold and $N$ the number of vertices in t…

Let $\mathcal G_{n,p}$ be the random graph model, with $p=p(n)$, and let $G,H$ be independent graphs sampled from $\mathcal G_{n,p}$. Let $G\mathbin{\square}H$ be their Cartesian p…

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…

A graph $G$ has the 2-pebbling property if two pebbles can be moved to any specified root from every configuration $C$ of size $2\pi(G)-s(C)+1$, where $s(C)$ is the number of verti…

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 $G$ be a graph, and let $D$ and $W$ be distributions on $G$. A value function is any function $A$ on the set of distributions satisfying the two properties of the standard valu…

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…

Fixed-$l$ grid-threshold conjecture. For fixed $l$,

High-connectivity Class 0 conjecture. Every graph of fixed diameter and high enough connectivity is Class 0.

Path-threshold transfer conjecture. There exists a graph sequence ${\cal G}$ such that

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 $w$ and $b>1$ be given, let ${\cal F}\in BMS[w,b]$, and define $x$ by … Let $\operatorname{shad}[{\cal F}]$ denote the shadow of ${\cal F}$. Multisubset shadow conjecture. Then…

Let ${\cal G}=\{G_1,\ldots,G_n,\ldots\}$ and ${\cal H}=\{H_1,\ldots,H_n,\ldots\}$ be graph sequences, and let $f(G_n)$ denote the pebbling number of $G_n$. Assume that…

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…

Strong hub cover pebbling conjecture for cycles.

Kneser graph Class 0 conjecture. Every Kneser graph is Class $0$.