Ore- and Pósa-type conditions for partitioning 2-edge-coloured graphs into monochromatic cycles
share
In 2019, 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. Continuing work by Allen, Böttcher, Lang, Skokan and Stein, we also show that if deg(u) + deg(v) ≥ 4n/3 + o(n) holds for all non-adjacent vertices u,v ∈ V(G), then all but o(n) vertices can be partitioned into three monochromatic cycles.
READ FULL TEXT