282 problems

Let $\mathcal{T}_n$ be the set of all trees with $n$ vertices, ordered by $\succeq$. The length of a chain is the number of elements in a sequence $T_1\succ T_2\succ\cdots$ under t…

Henning–Oellermann–Swart conjecture.

Veselovac's conjecture. The tree $T_h$ is stackable for all $h\ge 1$.

Let $T$ be a tree with diameter $d$, and let $m_T[0,1)$ denote the number of Laplacian eigenvalues of $T$ less than $1$. Almost-all-trees lower-bound conjecture. Almost all trees h…

Let $T$ be a tree of order $n$, let $d$ denote its diameter, and let $m_T[0,1)$ be the number of Laplacian eigenvalues of $T$ in the interval $[0,1)$. Define … where…

Let $k\geq 2$ and let $T$ be a tree. Write $\chi _{_{NL}}(T)$ for the neighbor-locating-chromatic number of $T$, and let $\Delta(T)$ denote its maximum degree. Maximum-degree conje…

Let $t$ be the number of leaves of a tree, and call a tree $t$-minimal if it has $t$ leaves and no other tree with the same number of leaves has a smaller ABC-index, where … for a…

Ilić–Stevanović's conjecture. The Volkmann tree has maximum spectral moment

Pemantle–Steif conjecture. If

Let $T_{a,b}=(A,B)$ be a tree whose bipartition satisfies … For a bipartite graph $H$, let $\operatorname{ex}_{\mathrm{bip}}(m,n;H)$ be the maximum number of edges in an $H$-free b…

Every tree is bipartite. Write $T_{a,b}=(A,B)$ when the two parts of $T$ satisfy … For a bipartite graph $H$, let $\operatorname{ex}_{\mathrm{bip}}(N;H)$ be the maximum number of e…

Let $T$ be a tree on $t$ vertices. For a graph $G$, write $e(G)$ for its number of edges, and call $G$ $T$-free if it contains no subgraph isomorphic to $T$. Erdős–Sós conjecture.…

Let $T$ be a tree with order $n$ and diameter $\operatorname{diam}(T)$, and let $md(T)$ denote its multiset dimension. Hafidh's conjecture. If $md(T)<\infty$, then … and this bound…

Let $T$ be an orientation of a tree with maximum degree at least $3$. An oriented graph $D$ is converse invariant if $f_T(D)=f_T(\overline{D})$ for every tournament $T$, where…

Let $D_1$ and $D_2$ be non-isomorphic oriented paths. Their chromatic noncommutative symmetric functions are denoted by $\mathbf{X}_{D_1}$ and $\mathbf{X}_{D_2}$, respectively. Ori…

Center–median conjecture. The center and median of $T_0$ have distance zero:

For $k\ge 1$, let $T$ be a tree of order $m$, and let $G$ be a $k$-connected or $k$-edge-connected graph with minimum degree $\delta(G)\ge k+m-1$. A subtree $T'$ is isomorphic to…

For $k\ge 1$ and any tree $T$ of order $m$, let $G$ be a $k$-connected graph with minimum degree $\delta(G)\ge \left\lfloor 3k/2\right\rfloor+m-1$. A subtree $T'$ is isomorphic to…

Subtree separation conjecture. The universal support-forest-counting combinations arising from $M_r^{(2)}(T)$, together with the Laplacian spectrum, determine all embedded forest c…

Spectral support-forest-profile conjecture. If $T_1$ and $T_2$ have the same Laplacian spectrum and the same $(n-2,2)$ support-forest profile, then $T_1$ and $T_2$ are isomorphic.…

Let $T$ be a tree on $n$ vertices. For a graph $G$ with no isolated vertices, define its Laplacian ratio by … where $L(G)$ is the Laplacian matrix, $d(v)$ is the degree of $v$, and…

Characterization problem. Characterize those trees $T$ for which

Let $T$ be a finite connected tree. Stacking–estimating conjecture. … The statement is presented as an immediate consequence of the preceding theorem under ASH, which gives the cor…

Let $i(T)$ denote the number of independent sets of a tree $T$. Effective Linek's Problem. Every integer greater than $88013$ appears as $i(T)$ for some tree $T$. This is an effect…

Tree form of the Burning Number Conjecture. Every tree on $n$ vertices satisfies