Decompositions of edge-colored infinite complete graphs into monochromatic paths

Elekes, Márton and Soukup, Dániel Tamás and Soukup, Lajos and Szentmiklóssy, Zoltán (2017) Decompositions of edge-colored infinite complete graphs into monochromatic paths. DISCRETE MATHEMATICS, 340 (8). pp. 2053-2069. ISSN 0012-365X

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Abstract

An . r . -edge coloring of a graph or hypergraph . G=(V,E) is a map . c:E→(0,...,r-1). Extending results of Rado and answering questions of Rado, Gyárfás and Sárközy we prove that . •the vertex set of every r-edge colored countably infinite complete k-uniform hypergraph can be partitioned into r monochromatic tight paths with distinct colors (a tight path in a k-uniform hypergraph is a sequence of distinct vertices such that every set of k consecutive vertices forms an edge);•for all natural numbers r and k there is a natural number M such that the vertex set of every r-edge colored countably infinite complete graph can be partitioned into M monochromatic kth powers of paths apart from a finite set (a kth power of a path is a sequence v0,v1,. of distinct vertices such that 1≤|i-j|≤k implies that vivj is an edge);•the vertex set of every 2-edge colored countably infinite complete graph can be partitioned into 4 monochromatic squares of paths, but not necessarily into 3;•the vertex set of every 2-edge colored complete graph on ω1 can be partitioned into 2 monochromatic paths with distinct colors. . . © 2016 Elsevier B.V.

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