9 problems

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

Let $G$ be a $d$-regular graph on $n$ vertices, and let $\rho(G)$ denote its path-cover number. The regular-graph path-cover conjecture. … The conjecture was proved in the cited so…

Double Hall path-cover conjecture. If

Let $G$ be a graph with maximum degree $\Delta$ and minimum degree $\delta$. Magnant et al.'s degree-based path-cover conjecture. The graph $G$ needs at most … paths to cover its v…

The Hall-type path-cover conjecture. Every $(X,Y)$-bigraph $G$ has a path $X$-cover by at most $\operatorname{{\mathsf\Lambda}-def}(G)$ paths.

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 $G$ be a block-cycle graph. A minimal path cover of $G$ is a path cover with the minimum possible number of paths, and a collection of forcing chains is the collection of direc…

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…