18 problems

Let $G$ be a 4-connected toroidal graph, meaning a $4$-connected graph embedded on the torus. The Grünbaum–Nash-Williams conjecture. $G$ is hamiltonian. The conjecture extends the…

Let $B_m(b)$ denote the toroidal chain of complete graphs with parameter $b$, and let $\lambda_{B_m(b),3,j}$ denote the corresponding eigenvalues indexed by $j$. Consider the four…

Let $G$ be a toroidal graph, meaning a graph embeddable on the torus. Let $u(G)$ denote its list chromatic number, written in the source as $c7ell(G)$. **Böhme et al.'s conjecture…

Let $M(G)$ be a toroidal cubic map whose 1-skeleton is a simple cubic graph of girth $4$. Call $G$ toroidally 3-edge connected when the bicutout condition defined in the source hol…

Improper coloring conjecture for toroidal graphs. Every toroidal graph is $(0,0,0,3)$-colorable.

Let $G$ be a toroidal graph containing no subgraph isomorphic to $K_5$ and no $6$-cycle. A graph is $4$-choosable if every assignment of lists of four colors to its vertices admits…

Uniqueness conjecture for the $K_{3,3}$ embedding. The proposed embedding of $K_{3,3}$, with coordinates as described in the table, is unique up to an isometry.

The spacing-five conjecture. There is no critical prism-canvas of spacing at least five.

The edge-width four conjecture. Every graph drawn on the torus with edge-width at least four is $5$-choosable.

The toroidal 5-choosability conjecture. A toroidal graph is $5$-choosable if and only if it is $5$-colorable.

Rectangular-flat-torus grid contact conjecture. Every bipartite toroidal graph without loops has a grid contact representation on the rectangular flat torus.

Rectangular-flat-torus tessellation conjecture. Every toroidal graph without loops has a tessellation representation on the rectangular flat torus.

Seven-colour conjecture for toroidal graphs. Every toroidal graph $G$ satisfies

Let $G$ be a planar or toroidal graph without intersecting triangles. Let $(H,f)$ be a cover of $G$, where $f$ assigns to each vertex of $H$ a value in $\{0,1,2\}$. A strictly $f$-…

Let $G$ be a toroidal graph with no subgraph isomorphic to any configuration in the family denoted by $E$. Let $(H,f)$ be a cover of $G$, where $f$ assigns to each vertex of $H$ a…

Let $T_{M^d}$ be the $d$-dimensional toroidal grid with all side lengths equal to $M$, and let $R_{\rm ave}(T_{M^d})$ denote its average effective resistance. More generally, let…

Cai–Wang–Zhu's conjecture. If $G$ has no $6$-cycles, then

Let $G$ be a graph embeddable in the torus. An arboreal 3-coloring of $G$ is a coloring with three colors such that each color class induces a forest. Exponential 3-arboral colorin…