6 problems

Double Hall path-cover conjecture. If

Magnant–Wang–Yuan's conjecture. The path-cover number satisfies

Magnant–Martin's strengthened conjecture. If $G$ is a $d$-regular digraph on $n$ vertices, then

Let $\vec{K}_{\mathbb{N}}$ be the complete symmetric digraph on the positive integers, and let a 2-edge-colouring assign one of two colours, red or blue, to every directed edge. A…

Erdős–Gyárfás same-color path cover conjecture. The vertex set of every $2$-colored $K_n$ can be covered by at most

Let $K_n$ be a complete graph whose edges are colored with three colors. A monochromatic path or cycle is a path or cycle all of whose edges have one color, and the two pieces are…