11 problems

Let $G$ be a 2-connected graph with edge set $E$, and let $P_G$ denote its symmetric edge polytope. For an edge $e\in E$, let $P_{G\setminus e}$ be the symmetric edge polytope of t…

Summed edge-deletion nonnegativity conjecture. The polynomial $Z_G(t)$ has nonnegative coefficients. This is proposed as a stronger statement whose gamma-nonnegativity would imply…

The preceding variance bound concerns triangulations in the regime where $p$ may approach $1$. Asymptotic equality conjecture. The result should hold with asymptotic equality, and…

Let $G$ be a 2-connected graph and let $ij$ be any edge of $G$. Let $\Gamma$ be a unimodular triangulation of the subcomplex of $P_{G\setminus ij}$ consisting of all facets visible…

Cross-degree and interlacing conjecture. The inequalities

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 $P$ be the symmetric edge polytope of a finite undirected graph, and write its Ehrhart $h^*$-polynomial in the form … The vector $(\gamma_0,\dots,\gamma_{\lfloor d/2\rfloor})$…

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…

Gamma-zero characterization conjecture. If $k\in\mathbb{N}$ and $G$ is a $k$-connected graph on $n$ vertices, then

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…