8 problems

Chen–Yang–Yuan–Zhang conjecture. For $n\geq 3k$ and $s\geq 3$,

Strong maximality characterization conjecture. The systems described in those two conjectures are exactly the strongly maximal systems of vertex-disjoint and internally vertex-disj…

Mader's internally-disjoint-path conjecture. Such a system $\mathcal{P}$, set $X$, and partition $\mathcal{Y}$ exist.

Gallai's conjecture. There exists such a system $\mathcal{P}$ and set $X$.

Lovász–Cherkassky conjecture. Under these hypotheses, there exists a system $\mathcal{P}$ of edge-disjoint $T$-paths having the required cut property for every $t\in T$.

Complementary-slackness conjecture. There exists a system $\mathcal{P}$ of edge-disjoint $T$-paths and a $T$-partition $\mathcal{A}$ satisfying those conditions.

Strong maximality conjecture. There exists a strongly maximal system $\mathcal{P}$ of edge-disjoint, vertex-disjoint, or internally vertex-disjoint $T$-paths in $G$.

Let $k$ be a positive integer, let $G$ be a graph of degeneracy $d$, and let $c_k$ denote a constant depending on $k$. A collection of paths is edge-disjoint if no two paths share…