20 problems

Path-power equitable list arboricity conjecture. The graph $G$ is equitably $k$-list arborable if and only if

Large-girth strong arboricity conjecture. For every integer $k$ there is an integer $g(k)$ such that every graph $G$ with

Let $G$ be a planar graph, and let $a_f(G)$ denote its fractional arboricity: the minimum ratio $p/q$ for which vertices can be assigned at least $q$ colors from a total of at most…

Let $G$ be a graph, let $a(G)$ denote its arboricity, and let $\textrm{linear }a(G)$ denote its linear arboricity, the minimum number of linear forests needed to cover the edges of…

Strong arboricity coloring-number conjecture. Every graph $G$ satisfies

Strong arboricity conjecture. Every graph $G$ satisfies

Planar strong arboricity conjecture. Every planar graph $G$ satisfies

Directed degree-$f$ arboricity conjecture. For every directed multigraph $D$,

Simple-graph degree-$f$ arboricity conjecture. Truszczyński's degree-$f$ arboricity conjecture should hold for simple graphs: for every such $G$ and $f$,

Truszczyński's conjecture. For every multigraph $G$ and function $f:V(G)\to\mathbb{Z}_{\ge 2}$, when $f$ is constant,

The asymptotic $H$-factor conjecture. Given $\mu>0$, $h\in\mathbb{N}$, and an $h$-vertex graph $H$ with $\mathrm{hcf}(H,\mathrm{AP})=1$, there exists $\alpha>0$ such that, for all…

Let $D$ be a digraph. Write $\gamma(D)$ for the fractional arboricity of its underlying graph, and let $\Delta^-(D)$ and $\Delta^+(D)$ denote its maximum in-degree and maximum out-…

Improved equitable list arboricity conjecture. Any connected graph $G$ is equitably $\lceil\Delta(G)/2\rceil$-list arborable provided $G$ is neither a cycle nor a complete graph of…

Separation conjecture. There is a graph $G$ and $k\in\mathbb{N}$ satisfying both properties above, but $G$ is not equitably $k$-list arborable.

Forest-and-bounded-diameter-forest decomposition conjecture. There exists a natural number $f(d)$ such that $G$ can be partitioned into two forests, each of whose components has di…

Bounded diameter arboricity plus-one conjecture. The class $\mathcal{A}_k$ has bounded diameter arboricity $k+1$, that is,

For graphs $H_1$ and $H_2$, let $r(H_1,H_2)$ be the least integer $n$ such that every red-blue edge-coloring of $K_n$ contains a red copy of $H_1$ or a blue copy of $H_2$. The arbo…

Let $G$ be a graph and let $k,d\in\mathbb{N}$. A vertex subset $A\subseteq V(G)$ is overfull if … A graph is $(k,d)$-sparse when it satisfies the corresponding $(k,d)$-sparseness i…

Let $G$ be a loopless multigraph. A weak $(k,d)$-decomposition is a decomposition of $G$ into $k$ forests and one $d$-bounded graph. Weak Nine Dragon Tree Conjecture. If … then $G$…

Montassier et al.'s conjecture. If