Partitioning two-coloured complete multipartite graphs into monochromatic paths

نویسندگان

Oliver Schaudt

Maya Jakobine Stein

چکیده

We show that any complete k-partite graph G on n vertices, with k ≥ 3, whose edges are two-coloured, can be covered by two vertex-disjoint monochromatic paths of distinct colours, under the necessary assumption that the largest partition class of G contains at most n/2 vertices. This extends known results for complete and complete bipartite graphs. Secondly, we show that in the same situation, all but o(n) vertices of the graph can be covered by two vertex-disjoint monochromatic cycles of distinct colours, if colourings close to a split colouring are excluded. The same holds for balanced complete bipartite graphs. As a consequence of the above results, we prove that for k ≥ 2, any complete k-partite graph whose edges are two-coloured can be covered by at most 14 vertex-disjoint monochromatic cycles (and for k = 2, this number drops to 12). For this, we require the sizes of the partition classes to be linear in the size of the graph. keywords: monochromatic path partition, monochromatic cycle partition, two-coloured graph

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