6 problems

Unimodality conjecture. The $h^*$-vector of $P_M(W_n)$ is unimodal for every $n\geq 4$.

Let $h_{R_n}(\boldsymbol{b})$ denote the Hilbert function counting lattice points in the fractional perfect $\boldsymbol{b}$-matching polytope of the complete graph $K_n$, equivale…

Wheel-graph degree conjecture. One has

Matching-polytope degree conjecture. One has

Let $\textit{PPerm}(m,n)$ denote the partial permutation polytope. Volume conjecture. The normalized volume of $\textit{PPerm}(n,2)$ (equivalently, $\textit{PPerm}(2,n)$) is equal…

Let $G$ be a graph, let $\hat{G}$ be its underlying simple graph, and let $\chi'(G)$ and $\chi'_f(G)$ denote its chromatic index and fractional chromatic index, respectively. The b…