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

Partitioning 3-Colored Complete Graphs into Three Monochromatic Cycles

We show in this paper that in every 3-coloring of the edges of K 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 cove...

Partitioning 3-coloured complete graphs into three monochromatic paths

Partitioning 2-edge-colored complete multipartite graphs into monochromatic cycles, paths and trees

In this paper we consider the problem of partitioning complete multipartite graphs with edges colored by 2 colors into the minimum number of vertex disjoint monochromatic cycles, paths and trees, respectively. For general graphs we simply address the decision version of these three problems the 2-PGMC, 2-PGMP and 2-PGMT problems, respectively. We show that both 2-PGMC and 2-PGMP problems are NP...

Partitioning edge-coloured complete graphs into monochromatic cycles and paths

A conjecture of Erdős, Gyárfás, and Pyber says that in any edge-colouring of a complete graph with r colours, it is possible to cover all the vertices with r vertexdisjoint monochromatic cycles. So far, this conjecture has been proven only for r = 2. In this paper we show that in fact this conjecture is false for all r ≥ 3. In contrast to this, we show that in any edge-colouring of a complete g...

Partitioning edge-2-colored graphs by monochromatic paths and cycles

We present results on partitioning the vertices of 2-edge-colored graphs into monochromatic paths and cycles. We prove asymptotically the two-color case of a conjecture of Sárközy: the vertex set of every 2-edge-colored graph can be partitioned into at most 2α(G) monochromatic cycles, where α(G) denotes the independence number of G. Another direction, emerged recently from a conjecture of Schel...