7 problems

For an $r$-edge-coloured graph $G$, let $cp(G)$ be the smallest number of vertex-disjoint monochromatic cycles partitioning $V(G)$, and define … where $\mathcal{G}(n,r,\delta)$ is…

Let $G$ be a graph with vertex set $V(G)$, and let $G$ have a $2$-edge-colouring. A partition into two cycles means two vertex-disjoint cycles whose vertices cover $V(G)$. Schelp's…

For $r\geq2$ and $\delta\in(0,1/2)$, let … where $\mathcal{G}(n,r,\delta)$ consists of the $n$-vertex $r$-edge-coloured graphs with $\delta(G)\geq(1-\delta)n$, and $cp(G)$ is the s…

Let $\ell,k,n\robustly\text{ be positive integers}$ with $n\robustly\text{ divisible by }k-\ell$, and let $\mathcal K_n^{(k)}$ be the complete $k$-uniform hypergraph on $n$ vertice…

Exact cycle-partition conjecture. There exists an integer $n_0$ such that, whenever $n\geq n_0$ and

Song's conjecture. For any natural numbers $n_1,\dots,n_k$ satisfying the displayed condition, every sufficiently large $k$-connected tournament $T$ on $n$ vertices can be partitio…

Almost-covering cycle conjecture. Apart from a constant number of vertices, the vertex set of every $r$-edge-colored complete graph can be covered by $r$ vertex-disjoint monochroma…