In 2019, Letzter confirmed a conjecture of Balogh, Bar\'at, Gerbner, Gy\'arf\'as and S\'ark\"ozy, proving that every large $2$-edge-coloured graph $G$ on $n$ vertices with minimum degree at least $3n/4$ can be partitioned into two monochromatic cycles of different colours. Here, we propose a weaker condition on the degree sequence of $G$ to also guarantee such a partition and prove an approximate version. Continuing work by Allen, B\"ottcher, Lang, Skokan and Stein, we also show that if $\operatorname{deg}(u) + \operatorname{deg}(v) \geq 4n/3 + o(n)$ holds for all non-adjacent vertices $u,v \in V(G)$, then all but $o(n)$ vertices can be partitioned into three monochromatic cycles.

翻译：2019年,Letzter确认了Balogh、Bar\'at、Gerbner、Gy\'arf\'as和S\'ark\'ozy的猜想,证明每张价值至少为30/40美元、价值至少为30/40美元的巨型顶色图形$G美元可以分割成两种不同颜色的单色循环。在这里,我们提议了一个以$G为单位的较弱条件,以保障这种分区并证明一个大致的版本。Allen、B\'ottcher、Lang、Skokkankan和Stein的继续工作,我们还表明如果$\opratorname{deg}(u)+\opratorname{deg}(v)\geq 4n/3+o(n)在所有非相形脊椎的美元V(G)之间可以分割成三个单色循环。