6 problems

Let $G$ be an $n$-vertex $r$-edge-coloured graph, and let $tc(G)$ be the smallest number of not necessarily vertex-disjoint monochromatic trees whose vertices cover $V(G)$. Bal–DeB…

For $r\geq2$ and $\delta\in(0,1)$, let … where $\mathcal{G}(n,r,\delta)$ consists of the $n$-vertex $r$-edge-coloured graphs with $\delta(G)\geq(1-\delta)n$, and $tc(G)$ is the sma…

Triangle-free tree cover conjecture. For all connected triangle-free graphs,

Let $K_{n,m}$ be a complete bipartite graph, and let $\operatorname{tc}_{r,1}(K_{n,m})$ denote the minimum number of monochromatic trees needed to cover every $r$-colouring of its…

Let $K_n$ be the complete graph on $n$ vertices. In an $(r,k)$-colouring, each edge receives a $k$-element subset of a set of $r$ colours; let $\operatorname{tc}_{r,k}(K_n)$ be the…

Let $t$ be a positive integer, let $s$ satisfy $1\le s\le t-1$, and let $K_n$ be a complete graph with its edges colored using $t$ colors. A monochromatic tree is a tree all of who…