13 problems

Let $\mathcal{F}$ be the barycentric subdivision, and let $H_{\mathcal{F}}$ be the transformation matrix encoding the transformation of the $h$-vector under an $\mathcal{F}$-unifor…

Let $\ell\geq 1$. A directed graph has minimum out-degree at least $3.23\ell$ if every vertex has at least $3.23\ell$ outgoing edges, and a $(2,\ell)$-spider is a $1$-subdivision o…

Spanning subdivision conjecture. There is a constant $n_0$ and a real number $C_H$ depending only on $H$ such that, whenever $D$ has order $n$ and the displayed condition holds for…

Pavez-Signé's conjecture. For all $C\geq C_0$ and any integer $m$, if $G$ is a graph on $n\geq Cm$ vertices with minimum degree at least $(1/2+\varepsilon)n$, then $G$ contains a s…

Mozafari-Nia–Iradmusa conjecture. For every simple graph $G$,

Let $\mathcal{F}$ be the barycentric subdivision, and let $H_{\mathcal{F}}$ denote its associated operator. A matrix is totally positive if every minor of every order is positive,…

Polynomial degeneracy conjecture for graphs excluding subdivisions. For every graph $H$, every $H$-subdivision-free graph $G$ that does not contain $K_{\ell,\ell}$ as a subgraph ha…

Let $s\geq2$, $k,b,t\geq1$ be integers. Let $S$ be an $s$-legged spider whose longest leg has length $k$, and suppose that $e(S)=(s-1)k+b$, where $0\leq b\leq k$. Let $t*S$ denote…

Let $X\subseteq\mathbb{R}^d$ be the sample configuration, let $X'\supseteq X$ and let $\Delta'$ be the finite set and bimonotone subdivision produced by the paper's algorithm. Let…

Chan–Stanley unimodality conjecture. All local $h$-vectors of quasi-geometric subdivisions are unimodal.

Athanasiadis's local gamma-vector conjecture. The local $\gamma$-vector of a flag vertex-induced homology subdivision of a simplex is nonnegative.

Let $C$ be an oriented cycle, and let a subdivision of $C$ be obtained by replacing each arc of $C$ by a directed path. A digraph is strong if every ordered pair of vertices is joi…

Let $\Delta$ and $\Delta'$ be flag homology spheres, and suppose that $\Delta'$ geometrically subdivides $\Delta$. Let $\gamma(\Delta)$ and $\gamma(\Delta')$ denote their gamma-vec…