Partitioning $3$-Colored Complete Graphs into Three Monochromatic Cycles

Abstract

We show in this paper that in every $3$-coloring of the edges of $K^n$ all but $o(n)$ of its vertices can be partitioned into three monochromatic cycles. From this, using our earlier results, actually it follows that we can partition all the vertices into at most 17 monochromatic cycles, improving the best known bounds. If the colors of the three monochromatic cycles must be different then one can cover $({3\over 4}-o(1))n$ vertices and this is close to best possible.