17 problems

Gyárfás–Sumner analogue for $\eta$-boundedness. For every forest $H$, there exists a function $f:\mathbb{N}\to\mathbb{N}$ such that every $H$-free graph $G$ satisfies…

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

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

Strong Nine Dragon Tree Conjecture. If

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…

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…

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

Kinnersley–West–Zamani 3/5-conjecture.

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…

Montassier et al.'s conjecture. If