18 problems

Let $D$ be a directed graph, and let $\operatorname{\mathcal{O}}(D)$ denote its ornamentation lattice. Polytopal realization conjecture. For any directed graph $D$, the ornamentati…

Let $G$ be a bipartite graph, and let $P$ be a facet of its symmetric edge polytope associated with an orientation of $G$. The standard orientation is the orientation in which ever…

Partition-graph $f$-polynomial conjecture. The approach developed in this paper may be modified to obtain the $f$-polynomials of partition graphs.

Bóna–Ju–Ann conjecture. If $k=2\ell$, then

Ju's conjecture. The polynomial in the numerator of this Ehrhart series is symmetric, also known as palindromic.

Higashitani–Kummer–Michałek conjecture. (i) For any complete multipartite graph $K_{a_1,\dots,a_k}$, the roots of $E_{a_1,\dots,a_k}$ lie on $\operatorname{CL}$. (ii) If…

Let $G$ be a connected graph with $n\geq 3$ vertices. Let ${\mathcal P}_G$ be its symmetric edge polytope, and let $N({\mathcal P}_G)$ denote the number of its facets. A $1$-sum of…

Full windmill maximizer conjecture. Among graphs with $n$ vertices and $3(n-1)/2$ edges, for odd $n$, $WM(n)$ is a facet maximizer. More generally, any graph that is a join of…

Coffee-bean maximizer conjecture. The facet count $N(CB(x_1,x_2,x_3))$ is maximized by

Different-parity coffee-bean conjecture. If $x_1$, $x_2$, and $x_3$ have different parities, then

Facet-count bounds conjecture. If $n=2k$ with $k\geq 2$, then $F(x_1,x_2,x_3)$ is maximized at $F(n-1,1,1)$. If $n=2k-1$ with $k\geq 3$, then it is maximized at $F(n-3,2,2)$. Furth…

Polytope characterization conjecture. There is a polytope $P$ whose vertices are labelled by the edges of $G$ and whose faces are in bijection with the induced subgraphs of $G$ hav…

For a graph $G$, let $\overline{G}$ be its complement, let $P_G$ be the associated graph polytope, and let $\mathsf{U}P_G$ denote the centrally symmetric polytope used in the paper…

Facet-orbit conjecture. If $G=M_{2m}$ with $m\ge 4$, then among the facets of $\operatorname{CUTP}(G)$ there are two orbits of $4m$ and $2m$ edge facets from the ring- and rung-edg…

Let $G$ be a complete $k$-partite graph of type $(a_1,\dots,a_k)$, and let $H_{a_1,\dots,a_k}$ denote its Ehrhart polynomial. Complete multipartite Ehrhart conjecture. The polynomi…

Let $G$ be the complete graph with $d$ vertices, let $\mathcal{P}^{\pm}_G$ be its symmetric edge polytope, and write $i(\mathcal{P}^{\pm}_G,m)$ for its Ehrhart polynomial. Complete…

Let $G$ be a complete bipartite graph of type $(2,d-2)$, with edges $\{1,j\}$ and $\{2,j\}$ for $3\leq j\leq d$, and let $\mathcal{P}^{\pm}_G$ be its symmetric edge polytope. Write…

Let $G$ be a finite connected simple graph, and let ${\mathcal P}_G$ be the convex hull of the vectors $\pm({\bold e}_i-{\bold e}_j)$ associated with the edges of $G$, viewed in th…