6 problems

Graham's tree reconstruction conjecture. For each sequence of natural numbers $a_0,a_1,a_2,\ldots$, all the conditions $|L^{(j)}(G)|=a_j$ for $j\geq 0$ are satisfied by at most one…

Let $T$ be a binary phylogenetic $X$-tree with $|X|=n$. For a character alignment $A_k(T)$ consisting of the binary characters of parsimony score $k$ on $T$, let…

Let $T$ be a binary phylogenetic $X$-tree with $|X|=n$. For a character alignment $A_k(T)$ consisting of the binary characters of parsimony score $k$ on $T$, let $k<\frac{n}{4}$. M…

Let $k\in\mathbb{N}_{\geq 3}$, let $X$ be a taxon set, and let $T$ and $\widetilde{T}$ be binary phylogenetic trees on $X$. For a binary phylogenetic tree $T$, let $A_k(T)$ denote…

Optimality conjecture. No distance-based algorithm has $l_\infty$-radius greater than $\frac{1}{4}$ for general trees. Consequently, if this assertion holds, the RNJ algorithm achi…

Projection-cone bound conjecture. Every maximal cell in $\mathcal{Q}_n$ is obtained as the intersection of at most $(n-1)!$ projection cones.