21 problems

Sub-leading Laurent coefficient conjecture. For every $d\ge5$ and every $m\ge2$,

Boundary-correction structure conjecture. For every $d\ge4$ and every $m,n$ with exactly one of them below $d-1$, the count $E_d(m,n)$ differs from $p_d(m,n)$ by a polynomial corre…

Let $E_d(m,n)$ be the count and let $p_d(m,n)$ denote its closed-form polynomial in the threshold regime. For $s=\min(m,n)<d-1$, define the boundary correction, with the other vari…

Let $d\ge5$, let $(a,b)$ satisfy $a+b=d$ and $a\le b$, and let $R_{(a,b)}(m,n)$ be the interior remainder. Denote by $[m^{d-3}]R_{(a,b)}(m,n)$ the coefficient of $m^{d-3}$; by symm…

Let $d\ge5$ and let $(a,b)$ be a split with $a\le b$. Write $R_{(a,b)}$ for the remainder after removing the single-side contributions from the count associated with the split. Deg…

Slowest-chain conjecture. Among all $\nu$-chains, the antidiagonal chain maximises

Let $S$ be a convex point set, and let $T_{\text{in}}$ and $T_{\text{tar}}$ be plane spanning trees on $S$. A parking edge is an edge appearing in a flip sequence that is not conta…

Let $S$ be a convex point set, and let $T_{\text{in}}$ and $T_{\text{tar}}$ be plane spanning trees on $S$. An edge is happy if it belongs to both $T_{\text{in}}$ and…

Let $\mathrm{WFG}_n$ be the wiggly flip graph. Hamiltonicity conjecture. The wiggly flip graph $\mathrm{WFG}_n$ admits a Hamiltonian path, or even a Hamiltonian cycle. The authors…

Let $P_{n+2}$ be a convex polygon, and consider its $2$-coloured flip graph, whose vertices are triangulations with triangles coloured using two colours. A triangle is understood b…

Let $\ell$ be a fixed even positive integer, and let $G(\underbrace{\ell,\dots,\ell}_{n\text{ copies}})$ denote the flip graph associated with the product of $n$ chains of length…

An $r$-signotope is a mapping from the $r$-subsets of an $n$-element set to $\\{+,-\\}$ satisfying the relevant monotonicity condition, and a flip changes the sign of a single $r$-…

Let $Q$ be a word and let $\omega$ be an element of a Coxeter group. For the non-empty subword complex $\mathcal{SC}(Q,\omega)$, let $[e,\omega]$ be the corresponding weak-order in…

Let $P$ range over all $3$-dimensional polytopes with $n$ vertices, and let $f$ range over all generic linear functionals on $P$. Write $G(P,f)$ for the flip graph associated with…

A flip replaces one of the two triangulations associated with a nearly $(n,k,m)$-admissible affine permutation by the other. Flip-connectivity conjecture. Any two triangulations of…

Triangle flip-graph connectivity conjecture. The triangle flip-graph on the set of all intersecting digon-free arrangements of $n$ pseudocircles is connected for every…

Let $P$ be a convex $n$-gon with $n\geq 13$, and let P^{\text{\raisebox{1px}{\scalebox{0.7}{star}}}} be the resulting polygon with a puncture placed in its interior. Write…

Let $Q$ and $Q'$ be quadrangulations related by twisting along one inner edge. Write $\gamma_Q$ and $\gamma_{Q'}$ for their undirected flip graphs, and $G_Q$ and $G_{Q'}$ for their…

Filling-surface growth-rate finiteness conjecture. The number of topological types of filling surfaces with the same growth rate is finite.

Unmarked-loop diameter growth conjecture. For every $\varepsilon>0$, there exists $k_\varepsilon$ such that, if $\Sigma$ is a surface with $k_\varepsilon$ unmarked boundary non-pri…

No-common-edge shortest paths conjecture. 1. A shortest path in $G_{n,k}$ between $T$ and $T'$ never flips a common edge of $T$ and $T'$. 2. If $f$ is a $k$-relevant edge of $T$,…