Minimum degree conditions for monochromatic cycle partitioning

Dániel Korándi, Richard Lang, Shoham Letzter, Alexey Pokrovskiy

Published 2019-02-15Version 1

A classical result of Erd\H{o}s, Gy\'ar\'as and Pyber states that any $r$-edge-coloured complete graph has a partition into $O(r^2 \log r)$ monochromatic cycles. Here we determine the minimum degree threshold for this property. More precisely, we show that there exists a constant c such that any $r$-edge-coloured graph on $n$ vertices with minimum degree at least $n/2 + c \cdot r \log n$ has a partition into $O(r^2)$ monochromatic cycles. We also provide constructions showing that the minimum degree condition and the number of cycles are essentially tight.