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Monochromatic cycle partitions in local edge colourings (1403.5975v3)
Published 24 Mar 2014 in math.CO
Abstract: An edge colouring of a graph is said to be an $r$-local colouring if the edges incident to any vertex are coloured with at most $r$ colours. Generalising a result of Bessy and Thomass\'e, we prove that the vertex set of any $2$-locally coloured complete graph may be partitioned into two disjoint monochromatic cycles of different colours. Moreover, for any natural number $r$, we show that the vertex set of any $r$-locally coloured complete graph may be partitioned into $O(r2 \log r)$ disjoint monochromatic cycles. This generalises a result of Erd\H{o}s, Gy\'arf\'as and Pyber.