Approximate Erdős–Gyárfás–Pyber cycle covering conjecture

Let $K_n$ be a complete graph whose edges are coloured with $r$ colours. A vertex-disjoint monochromatic cycle packing is a collection of $r$ vertex-disjoint monochromatic cycles. Approximate cycle covering conjecture. For each $r$ there is a constant $c_r$, such that in every $r$-edge-coloured complete graph $K_n$, there are $r$ vertex-disjoint monochromatic cycles covering $n-c_r$ vertices of $K_n$. The conjecture is open for $r\geq 3$; for $r=3$, the paper proves a weaker asymptotic version in which $c_r$ is replaced by a function $o_r(n)$ satisfying $o_r(n)/n\to0$ as $n\to\infty$.