30 problems

Let $G=G(\mathbb{F}_2^r,M)$ be a connected Cayley graph defined by a matroid $M$, with generator vectors $v_1,\ldots,v_n$, and let $d(M)$ denote the number of Sylow-$2$ cyclic fact…

Bipartiteness conjecture. $G$ is bipartite.

Sandpile group distribution conjecture. For every finite abelian group $G$,

Let $G_{\alpha}(n,u)=G(n,\lceil\alpha n\rceil,u)$ be the Erdős–Rényi random bipartite graph with edge probability $u$, let $S_{G_{\alpha}(n,u)}$ be its sandpile group, and let…

Let $n_1\geq n_2$ be positive integers, let $0<u<1$, and let $G(n_1,n_2,u)$ be the Erdős–Rényi random bipartite graph whose bipartition has vertex sets of sizes $n_1$ and $n_2$, wi…

Rank distribution conjecture. If $p$ is odd, then

Sylow distribution conjecture. Then

Bhargava–DePascale–Koenig conjecture. For constants $0<v<1$, $\frac{1}{p}<\alpha\leq 1$, and a finite abelian $p$-group $G$,

Let $M$ be an oriented regular matroid, equipped with triangulating circuit and cocircuit signatures. A consistent sandpile torsor algorithm is a consistent algorithm assigning san…

A plane graph is a planar graph equipped with a planar embedding, and a sandpile torsor algorithm assigns a sandpile torsor action to each plane graph. The rotor-routing and Bernar…

Let $T$ and $T'$ be trees, and let $T<T'$ in Csikvári's poset on trees. For a tree $S$, write $T_{\operatorname{Cone}(S)}(1,y)$ for the Tutte polynomial of the cone over $S$, evalu…

Let $M=(E,\mathcal B,\chi)$ be an oriented regular matroid, let $(\sigma,\sigma^*)$ be a pair of acyclic signatures, and let $\alpha$ and $\beta$ be matroidal sandpile torsor algor…

Let $M=(E,\mathcal B,\chi)$ be an oriented regular matroid, let $(\sigma,\sigma^*)$ be a pair of acyclic circuit and cocircuit signatures, and let $\alpha_{(M,\sigma,\sigma^*)}$ be…

Let $(G,\chi)$ be a plane graph, meaning a planar graph equipped with a plane ribbon structure. A sandpile torsor structure assigns a free transitive action of the sandpile group…

Let $p$ be a prime, and consider a random balanced bipartite graph, meaning that its two vertex classes have the same size. Let the $p$-rank denote the rank of the graph's sandpile…

Path-product conjecture. The only graphs $P_i\square P_j$ with the complete maximal identity property are $P_2\square P_j$ when $j\equiv1\pmod{3}$, together with $P_2\square P_2$,…

Let $T$ be a tree, let $deg(T)$ denote its degree as used in the source, and let $G_{T,c}$ be the associated graph with parameter $c$. Write ${\bf 1}$ for the all-ones configuratio…

Let $G$ be a finite graph, and let $K(G)$ denote its sandpile group. A finite group is cyclic if it can be generated by one element. Cyclicity conjecture. Almost all graphs have cy…

For $n\geq 1$, let $Q_m$ denote the $m$-dimensional hypercube and let $\operatorname{Syl}_2(K(Q_m))$ denote the Sylow-$2$ component of its sandpile group. Doubling conjecture. … Eq…

Let two Cayley graphs have generator multiplicities, and let $GL$-equivalence mean equivalence under the natural action of the relevant general linear group on the generators. GL-e…

Let $M$ be a generating multiset for a Cayley graph, and suppose the greatest common divisor of all generator multiplicities in $M$ is $1$. Eigenvalue-determination conjecture. The…

Let $M$ define a connected Cayley graph on $\mathbb{F}_2^r$, let $d(M)$ be the number of Sylow-$2$ cyclic factors in its sandpile group, and let the graph's eigenvalues be those as…

Let $G$ be a bi-connected outerplanar graph. Let $G^*$ be its inner dual, and let $P_1\cup\cdots\cup P_t$ be the polygon chain decomposition of $G$ corresponding to the maximal pat…

Let $G$ be an Erdős–Rényi random graph, and let $S(G)$ denote its sandpile group. A finite abelian group is cyclic when it can be generated by one element. Clancy's conjecture. Bas…

Let $G$ be an Erdős–Rényi random graph, let $\Gamma(G)$ be its sandpile group, and let $\langle\cdot,\cdot\rangle_G$ denote its canonical symmetric perfect bilinear pairing. The pa…