28 problems

Strong Nine Dragon Tree Conjecture. If

Let $T$ be a forest, let $I(T)$ be its edge ideal, and let a tree-like system for $I(T)$ be a system whose support is the set of edge monomials of $I(T)$. For stretched forests, th…

Let $R$ be a polynomial ring over a field whose indeterminates are the vertices of a forest $G$, and let $I(G)$ be the edge ideal generated by the squarefree quadratic monomials co…

Linear Erdős–Pósa conjecture. At least one of the following holds:

KAMAK tree conjecture. Every grounded forest is $\delta^+$-enforcible.

Sparse restricted forest-coloring conjecture. Every $(\frac{9}{5},\frac{2}{5})$-sparse simple graph has an $(\mathcal{F}^{\Delta}_{1},\mathcal{F})$-coloring.

Let $G$ be a forest, let $I(G)$ be its edge ideal in the polynomial ring $R=\mathbb{K}[x\mid x\in V(G)]$, and let $\nu(G)$ denote its matching number. For $1\leq k\leq\nu(G)$, let…

Let $F$ be a forest on $n$ vertices, and let $\chi'_{\cup}(F)$ denote its union vertex-distinguishing edge chromatic number, where edge colors are sets and the union of the colors…

Forest regularity conjecture. For every $1\leq k\leq \operatorname{mat}(G)$,

Mader's forest-orientation conjecture. Every orientation of a forest is $\delta^+$-maderian.

Let $\varepsilon>0$. A graph is globally $m$-bounded if no colour is used on more than $m$ edges, and a forest of order $k$ has $k$ edges. Rainbow forest packing conjecture. There…

Bipartite forest conjecture. For every forest $T$ there exists $\epsilon>0$ such that either some vertex of $G$ has degree at least $\epsilon n$, or there are anticomplete sets…

Incoherence conjecture for forests. The ideal $\mathcal I$ is incoherent if and only if $H$ is a forest.

Let $H$ and $H'$ be ordered forests, and let $R_<(H,H')$ denote the family of Ramsey graphs of $(H,H')$. Forest Ramsey-graph conjecture. If $(H,H')$ is Ramsey finite, then…

Let $H$ and $H'$ be ordered forests, and let $t$ be a positive integer. Write $R_<(H,H')$ for the family of Ramsey graphs of $(H,H')$, and let $\operatorname{girth}(F)$ denote the…

Let a constellation be a forest whose components are stars, and call it odd when it consists of an odd number of stars. A labeling of a graph is super edge-magic if it is an edge-m…

Caro, Lauri, and Zarb's conjecture. If $F$ has order at most

Let $F$ be a forest on $n$ vertices, and let $f(F)=f_2(F)$ denote the minimum number of vertices that must be deleted to obtain an induced subgraph with at least two vertices attai…

Andreae's conjecture. No countable locally finite forest is non-edge-reconstructible.

Let $G$ be an isolate-free graph on $n$ vertices. The D-game domination number $\gamma_g(G)$ is the total number of vertices chosen when Dominator starts and both players play opti…

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 $G$…

Subforest preservation conjecture. There exists a subforest $F'\subseteq F$ such that

Kinnersley–West–Zamani 3/5-conjecture.

Let $F$ be a forest and let $S$ be a star forest on the same vertex set. Sauer's conjecture. The union $F\cup S$ is always $3$-colorable. Stiebitz verified this conjecture, so the…

Let $(F^{c(n)}_n)_{n geq 0}$ be a sequence of forests, where $F^{c(n)}_n$ has $n$ vertices and $c(n)$ components. Let $X^{c(n)}_n$ be the number of classes in a uniformly chosen pa…