16 problems

Hilton–Zhao conjecture. If $\Delta(G_{\Delta})\leq 2$, then $G$ is overfull.

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

Kahn's conjecture. As the relevant degree parameters tend to infinity, the chromatic index of a $k$-uniform hypergraph is asymptotically equivalent to its fractional chromatic inde…

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…

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

Generalized Vizing's conjecture. Every linear hypergraph $\mathcal{H}$ without loops 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…

Let $G$ be a graph, and let $\chi'_k(G)$ denote its mod $k$ chromatic index. Bounded additive-gap conjecture. There is a constant $C$ such that … for every graph $G$. This conjectu…

Goldberg–Gupta–Seymour conjecture. For any graph $G$, if

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…

Let $G$ be a connected simple graph. Write $\Delta$ for its maximum degree, let $G_\Delta$ be the subgraph induced by the vertices of degree $\Delta$, and let $\chi'(G)$ be its chr…

Elementary-multigraph conjecture. Every multigraph $G$ with

Andersen–Seymour conjecture.

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 simple hypergraph on $n$ vertices, meaning that any two hyperedges share at most one vertex. Let $\chi_e(G)$ denote the chromatic index of $G$, the minimum number of c…