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Lehel conjectured that in every $2$-coloring of the edges of $K_n$, there isa vertex disjoint red and blue cycle which span $V(K_n)$. \L uczak, R\"odl, andSzemer\'edi proved Lehel's conjecture for large $n$, Allen gave a differentproof for large $n$, and finally Bessy and Thomass\'e gave a proof for all $n$.  Balogh, Bar\'at, Gerbner, Gy\'arf\'as, and S\'ark\"ozy proposed a significantstrengthening of Lehel's conjecture where $K_n$ is replaced by any graph $G$with $\delta(G)> 3n/4$; if true, this minimum degree condition is essentiallybest possible. We prove that their conjecture holds when$\delta(G)>(3/4+o(1))n$. Our proof uses Szemer\'edi's regularity lemma alongwith the absorbing method of R\"odl, Ruci\'nski, and Szemer\'edi by firstshowing that the graph can be covered with monochromatic subgraphs havingcertain robust expansion properties.

Monochromatic cycle partitions of graphs with large minimum degree