29 problems

Let $K$ be a knot, and let $c_n(K)$ denote its $n$-crossing number, the minimum number of $n$-fold crossings in a projection of $K$. Strict inequality conjecture. For all knots $K$…

Let $P$ be a generic set of $n$ points in the plane. The linear lower-bound conjecture. … The authors identify this as the most important problem for improving the lower bound for…

Hill's conjecture. The crossing number satisfies

Let $K_n^4$ denote the complete balanced $4$-partite graph with $n$ vertices, and let $\operatorname{cr}(G)$ and $\overline{\operatorname{cr}}(G)$ denote respectively its ordinary…

Let $K_{m,n}$ be the complete bipartite graph with part sizes $m$ and $n$, and define … Here $\operatorname{cr}(G)$ denotes the minimum number of crossings in a plane drawing of a…

Let $K_n$ be the complete graph on $n$ vertices, and define … Here $\operatorname{cr}(G)$ denotes the minimum number of crossings in a plane drawing of a graph $G$. Harary–Hill con…

Nonattainment conjecture. There exists no $f\in\mathcal F$ such that

Lamm's conjecture. $c_t(K)=c(K)$ if and only if $K$ can be represented by a continued fraction expansion $[a_1,\ldots,a_n]$ of type (A) or type (B).

Let $G_1$ and $G_2$ be graphs embedded on a closed surface $S$, and let the joint crossing number be the minimum number of crossing points between $h(G_1)$ and $G_2$ over all homeo…

Let $M_{n,t}$ be the graph considered in the construction above, and write $k=\lfloor n/2\rfloor$. If a set $P'\subseteq P$ has strength $0$ and $t=|P\setminus P'|$, then the const…

Let $G$ be a graph, and for a surface $[200~\Sigma$ let $\mathop{\rm iocr}\nolimits_{\Sigma}(G)$ denote the fewest number of pairs of independent edges that cross oddly in a drawin…

Let $cr(K_{n,n})$ be the crossing number of the complete bipartite graph $K_{n,n}$, and let $cr_0(K_{n,n})$ be its geodesic crossing number. Define … and … The limits exist, and sa…

Let $\mathcal{S}_n$ be the Sunlet graph on $2n$ vertices, obtained by attaching $n$ pendant edges to the cycle $C_n$, and let $K_{1,m}$ be the star graph on $m+1$ vertices. For a g…

Spider lower-bound conjecture.

For a drawing $D$ of $K_n$, define its deficiency by … The drawing $D$ has the natural deficiency property if, for every vertex $v$ of $K_n$, … Here a convex drawing is a drawing i…

Let $Q_8$ be the 8-dimensional hypercube, and let $cr_2(Q_8)$ denote its biplanar crossing number, the minimum total number of crossings in a drawing of $Q_8$ on two planes. Biplan…

Crossing-number equality conjecture.

Rectilinear crossing-number conjecture.

Let $T$ be a tile, and let $c_n(T)=\operatorname{cr}(\bigcirc(T^n))$. The limit … exists. Rationality conjecture for crossing limits. There exists a computable function $f$ that as…

Kanenobu knots crossing-number upper-bound conjecture. The crossing number satisfies

Let $D$ be a simple drawing of a graph, and let $E_{{\le}{\le}{\le}k}(D)$ denote the number of its $\le\!\le\!\le k$-edges, with … Here $E_i(D)$ denotes the number of $i$-edges in…

Let $D$ be a simple drawing of a graph, and let $E_{{\le}{\le}{\le}k}(D)$ denote the number of its $\le\!\le\!\le k$-edges, defined by … Here $E_i(D)$ denotes the number of $i$-edg…

Let $K_{a_1,a_2,a_3,n}$ denote the complete multipartite graph whose parts have sizes $a_1,a_2,a_3,n$, let $K_{2,4,n}$ denote the complete multipartite graph with parts of sizes…

Crossing-number conjecture.

Černý–Kynčl–Tóth dense stability conjecture. There exists an $\overline{\epsilon}\in(1/3,1)$ such that, for each $\epsilon\in(\overline{\epsilon},1]$, the family of graphs with…