Partitioning Edge-Colored Hypergraphs into Few Monochromatic Tight Cycles

Confirming a conjecture of Gy´arf´as, we prove that, for all natural numbers k and r, the vertices of every r-edge-colored complete k-uniform hypergraph can be partitioned into a bounded number (independent of the size of the hypergraph) of monochromatic tight cycles. We further prove that, for all natural numbers p and r, the vertices of every r-edge-colored complete graph can be partitioned into a bounded number of pth powers of cycles, settling a problem of Elekes, Soukup, Soukup, and Szentmikl´ossy [Discrete Math., 340 (2017), pp. 2053–2069]. In fact we prove a common generalization of both theorems which further extends these results to all host hypergraphs of bounded independence number

Partitioning edge-colored hypergraphs into few monochromatic tight cycles

Confirming a conjecture of Gyárfás, we prove that, for all natural numbers k and r, the vertices of every r-edge-colored complete k-uniform hypergraph can be partitioned into a bounded number (independent of the size of the hypergraph) of monochromatic tight cycles. We further prove that, for all natural numbers p and r, the vertices of every r-edge-colored complete graph can be partitioned into a bounded number of pth powers of cycles, settling a problem of Elekes, Soukup, Soukup, and Szentmiklóssy [Discrete Math., 340 (2017), pp. 2053-2069]. In fact we prove a common generalization of both theorems which further extends these results to all host hypergraphs of bounded independence number