26 problems

Let $G$ be a simple graph, let $A_G$ be its incidence matrix, let $I_G$ be the corresponding toric ideal, and let $M$ be a minimal Markov basis for $I_G$. A Markov basis is distanc…

Distance-irreducible basis conjecture. If $M$ is not distance reducing, then

Let $C$ denote the set of small conditionals, let $J-1$ be the dimension of the underlying lattice, and consider the coefficient matrix of the Diophantine equation governing the ma…

Bounded Markov complexity conjecture. If $\{1,2,\ldots,j\}$ is an independent subset of the underlying graph of $\Delta$, then the preceding bounded-format conclusion holds. This w…

Edge-deletion monotonicity conjecture. If $H$ is obtained from $G$ by deleting an edge, then

Complete bipartite graph ideal degree conjecture. The graph ideal for $G=K_{m,n}$ is generated in degree at most

Tree-width conjecture. The invariant $mu(G)$ is a function only of the tree width of $G$.

Planar graph Markov width conjecture. There is a universal constant $C$ such that

Distance-reducing inclusion conjecture. For every matrix $A$,

Closure conjecture. The set $\mathcal A$ is closed under gluing: if $A,B\in\mathcal A$ admit a gluing, then

Let $A$ be the incidence matrix of the complete graph $K_5$ without self-loops, namely … Let ${\mathbb N}A$ denote the semigroup generated by the columns of $A$, and for each…

Uniform Markov degree conjecture. The maximum Markov degree of the displayed iterated toric fiber product is bounded by a constant independent of $r_1,\dots,r_s$. This general conj…

Bounded-degree conjecture. The assumption that all factors arising in the toric fiber product have normal semigroups is unnecessary: there is a function $C(d_1,d_2)\in\mathbb{N}$ s…

Let $M$ be the matrix in the equation defining the space of tables, and let $A$, $B$, and $C$ denote the index sets used to specify the table and its conditionals. A total Markov b…

Width-and-degree generation conjecture. From the computation, one conjectures that $\ker(\pi)$ is generated, up to symmetry, by moves of width and degree at most three.

Let $A^{T}$ be the design matrix of the toric homogeneous Markov chain model without loops and initial parameters, with $S=3$ states and $T\geq 3$. Let $I_{A^{T}}$ denote its toric…

Let $C$ be a cycle of length $n$, with edges $12, 23, \ldots, n1$. For a vertex $i$, let $d_i$ denote its associated model parameter, and let $\mu(C,d)$ be the Markov width of the…

Let $\Gamma$ be a triangulation of a sphere of dimension $n$. The Markov basis of $I_{\Gamma, {\bf 2}}$ is a finite set of binomials associated to the simplicial complex $\Gamma$.…

Bounded-loop conjecture. For each $S$, there exists an integer $k_S$ such that a Markov basis consists of moves that are sums of at most $k_S$ extended simple loops.

Lifting and overlapping conjecture. Minimal Markov and Gröbner bases for the $p_1$ models can be obtained from Markov and Gröbner bases of $I_{\mathcal A_n}$ by repeated lifting an…

Minimal generation conjecture for the edge-reciprocation model. For $n\geq4$, the ideal $I_{\tilde{\mathcal E}_n}$ is minimally generated by homogeneous binomials of degrees $2$ an…

Degree conjecture for the constant-reciprocation model. The ideal $I_{\tilde{\mathcal C}_n}$ is generated in degrees $2$, $3$, and $4$, and the binomials are of the forms described…

Consider two-way contingency tables under the independence model, with some cell counts subject to upper bounds and other cell counts unbounded. In the unbounded case, the Markov b…

Let $m$ covariates index combinations of levels, and let ${\cal B}_{\Lambda(A_1 \otimes \dots \otimes A_m)}$ be the set of moves … Consider the subset for which every element of…

Let $m$ covariates index combinations of levels, and let $\Bj=(j_1,\dots,j_m)$ denote such a combination. Let $\bm{e}_{\Bj}$ be the array with $1$ at cell $(1,\Bj)$ and $-1$ at cel…