11 problems

Alon–Kim conjecture. For every $k\ge t\ge 1$ and $\varepsilon>0$, there exists $D_0$ such that, for every $D\ge D_0$, every $k$-uniform, $t$-simple hypergraph with maximum degree a…

Overfull Conjecture. Let $G$ be a graph of Class $2$ with

Berge–Füredi conjecture. A linear (loopless) hypergraph $(V,E)$ satisfies

Let $G$ be a simple graph and let $k\geq 2$ be an integer. A $\chi'_{k}$-coloring of $G$ is an edge coloring such that the subgraph induced by the edges of each color has all degre…

Goldberg's conjecture. If

Let $G$ be a simple connected graph with maximum degree $\Delta\ge 3$. The core $G_\Delta$ is the subgraph induced by the vertices of degree $\Delta$, and $\Delta(G_\Delta)\le 2$.…

Let $G$ be an HZ-graph, meaning a connected Class 2 graph with $\Delta(G_\Delta)\le2$, and let $P^*$ be the Petersen graph with one vertex removed. Let $\mathcal{O}_\Delta$ be the…

Let $G$ be a graph with vertex set $V(G)$, maximum degree $\Delta$, chromatic index $\chi'(G)$, and fractional chromatic index $\chi'_f(G)$. Hilton's Overfull Conjecture. If … then…

Elementary-multigraph conjecture. Every multigraph $G$ with

The $r$-adjacent strong chromatic index conjecture. For each positive integer $r$ there exist constants $\delta_0$ and $C$ such that

Let $G$ be a finite simple graph with no isolated vertices. Write $V(G)$ and $E(G)$ for its vertex and edge sets, let $\Delta(G)$ be its maximum degree, and let $\chi'_{a}(G)$ deno…