8 problems

Gyárfás–Lehel conjecture. In every $r$-coloring of the edges of $G$, the vertex set can be covered by the vertices of at most $2r-2$ monochromatic components.

Fix $k\geq 3$. Let $G$ be a graph with $n$ vertices and minimum degree satisfying … If the edges of $G$ are $k$-coloured, meaning that each edge receives one of $k$ colours, then G…

Gyárfás's conjecture. For fixed $k$, there exist sets $A_1,A_2,\dots,A_{k-1}$ whose union is $[n]$, and colours $c_1,c_2,\dots,c_{k-1}$, such that $K_n[A_i,c_i]$ is connected for e…

Generalized covering conjecture.

The complete-partite covering conjecture.

Distinct-color conjecture. If

Random-graph tree-cover threshold conjecture. For every $\epsilon>0$ and every $r\geq 1$, if

Ryser–Lovász conjecture. For every integer $r\geq 2$ and every graph $G$,