Partitioning complete bipartite graphs by monochromatic cycles (Q1354726)

For every positive integer \(r\) there exists a constant \(C_r\) depending only on \(r\) such that for every colouring of the edges of the complete bipartite graph \(K^{n,n}\) with \(r\) colours, there exists a set of at most \(C_r\) monochromatic cycles whose vertex sets partition the vertex set of \(K^{n,n}\). This answers a question raised by \textit{P. Erdös}, \textit{A. Gyárfás} and \textit{L. Pyber} [J. Comb. Theory, Ser. B 51, No. 1, 90-95 (1991; Zbl 0766.05062)].