12 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.

Gyárfás–Kiraly conjecture. For all $t\geq 1$ and $r\geq 3$,

Let $K_n$ be the complete graph on $n$ vertices, and let $\tau_r(G)$ denote the minimum integer $m$ such that every $r$-colouring of the edges of $G$ has a collection of $m$ monoch…

Let $t\geq 2$, and let $\gamma(t)$ denote the largest value such that every $(t+1)$-edge-coloring of every $n$-vertex graph with minimum degree at least $(1-\gamma(t))n$ contains a…

Given a graph $G$ and a positive integer $r$, let $\operatorname{mc}_r(G)$ be the largest integer $m$ such that every $r$-edge-coloring of $G$ contains a monochromatic component of…

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…

Grytczuk–Yeo–Szymański conjecture. If

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

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