27 problems

List Colouring Conjecture. For every graph $G$,

Defective list edge-colouring conjecture. For every graph $G$ and every integer $d$,

Let $G$ be a finite, undirected, loopless graph with no parallel edges, and let $\Delta$ be its maximum degree. A proper edge colouring $c$ of $G$ is adjacent vertex distinguishing…

Füredi–Kahn–Seymour conjecture. Then

Odd-defect list edge-colouring conjecture. For every odd integer $d$ and for every graph $G$,

Cambie's conjecture. Then

3-edge-colouring conjecture. Every cubic Cayley graph admits a 3-edge colouring.

Let $K_{2n}$ be the complete graph on $2n$ vertices, and let $L$ be a random $(p,2n-1)$-list assignment, in which each colour from $[2n-1]$ is included independently in each edge's…

Let $K_{n,n}$ be the complete bipartite graph with parts of size $n$. A random $(k,n)$-list assignment assigns independently and uniformly to each edge a $k$-element subset of…

Defective Goldberg-Seymour conjecture. Every graph $G$ satisfies

Grünbaum's conjecture. Every cubic graph with a polyhedral embedding in an orientable surface has an edge-3-colouring; equivalently, there are no cubic graphs with chromatic index…

Local Vizing's theorem. There is an $L$-edge-colouring of $G$.

List Edge Colouring Conjecture.

Bryant's conjecture. There exists a $K_3$-decomposition of $L\vee K_w$ if and only if all the following conditions hold: $\deg_L(x)\equiv w\pmod{2}$ for every vertex $x$ of $L$;…

Let $G$ be a connected graph with minimum degree $delta\geq 2$ and maximum degree $\Delta$, and let $\chi'_{\subset}(G)$ denote its inclusion chromatic index, the least number of c…

Let $G$ be a graph of maximum degree $\Delta$, and assign to each edge a list of available colours. A proper list edge colouring chooses for every edge a colour from its list, with…

For a graph $G$, let $\chi'(G)$ denote its edge chromatic number and let $\operatorname{ch}'(G)$ denote its list chromatic index. List colouring conjecture. For all graphs $G$, … T…

Let $G$ be a multigraph with maximum degree $\Delta(G)$ and maximum multiplicity $\mu(G)$. A forbidden matching is a matching whose edges have been assigned arbitrary, not necessar…

Let $G$ be a multigraph with maximum degree $\Delta(G)$ and maximum multiplicity $\mu(G)$. A precoloured matching is a matching whose edges have been assigned colours from a palett…

Let $G$ be a graph, let $k\in\mathbb{Z}^+$, and let $D$ be a $k$-optimal set, meaning a $k$-dependent set maximizing $\phi_k(D)=k|D|-|E(G[D])|$. A subgraph is $k$-edge-chromatic if…

Let $H$ be a hypergraph of $d$-intervals. Let $\chi_e(H)$ denote its edge chromatic number, and let $\Delta$ be the maximum degree of a point on any line. Edge-colouring conjecture…

Harary–Plantholt's conjecture. For every graph $G$,

Horňák–Woźniak's conjecture. For every graph $G$,

Zhang–Chen–Li–Yao–Lu–Wang's conjecture. For every graph $G$,

Baril–Togni's conjecture. If $G\neq C_5$, then