Ore- and Pósa-type conditions for partitioning 2-edge-coloured graphs into monochromatic cycles (Q6046228)

Summary: In [Electron. J. Comb. 26, No. 1, Research Paper P1.19, 67 p. (2019; Zbl 1406.05067)], \textit{S. Letzter} confirmed a conjecture of Balogh, Barát, Gerbner, Gyárfás and Sárközy, 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. This resembles a similar generalisation to an Ore-type condition achieved by \textit{J. Barát} and \textit{G. N. Sárközy} [J. Graph Theory 81, No. 4, 317--328 (2016; Zbl 1333.05235)]. Continuing work by \textit{P. Allen} et al. [``Partitioning a 2-edge-coloured graph of minimum degree \(2n/3 +o(n)\) into three monochromatic cycles'', Preprint, \url{arXiv:2204.00496}], we also show that if \(\operatorname{deg}(u) + \operatorname{deg}(v) \geqslant 4n/3 + o(n)\) holds for all non-adjacent vertices \(u,v \in V(G)\), then all but \(o(n)\) vertices can be partitioned into three monochromatic cycles.