83 research outputs found

    Vertex covers by monochromatic pieces - A survey of results and problems

    Get PDF
    This survey is devoted to problems and results concerning covering the vertices of edge colored graphs or hypergraphs with monochromatic paths, cycles and other objects. It is an expanded version of the talk with the same title at the Seventh Cracow Conference on Graph Theory, held in Rytro in September 14-19, 2014.Comment: Discrete Mathematics, 201

    Decompositions of edge-colored infinite complete graphs into monochromatic paths

    Get PDF
    An rr-edge coloring of a graph or hypergraph G=(V,E)G=(V,E) is a map c:E→{0,…,r−1}c:E\to \{0, \dots, r-1\}. Extending results of Rado and answering questions of Rado, Gy\'arf\'as and S\'ark\"ozy we prove that (1.) the vertex set of every rr-edge colored countably infinite complete kk-uniform hypergraph can be partitioned into rr monochromatic tight paths with distinct colors (a tight path in a kk-uniform hypergraph is a sequence of distinct vertices such that every set of kk consecutive vertices forms an edge), (2.) for all natural numbers rr and kk there is a natural number MM such that the vertex set of every rr-edge colored countably infinite complete graph can be partitioned into MM monochromatic kthk^{th} powers of paths apart from a finite set (a kthk^{th} power of a path is a sequence v0,v1,…v_0, v_1, \dots of distinct vertices such that 1≤∣i−j∣≤k1\le|i-j| \le k implies that vivjv_iv_j is an edge), (3.) the vertex set of every 22-edge colored countably infinite complete graph can be partitioned into 44 monochromatic squares of paths, but not necessarily into 33, (4.) the vertex set of every 22-edge colored complete graph on ω1\omega_1 can be partitioned into 22 monochromatic paths with distinct colors

    Partitioning infinite hypergraphs into few monochromatic Berge-paths

    Get PDF
    Extending a result of Rado to hypergraphs, we prove that for all s, k, t∈ N with k≥ t≥ 2 , the vertices of every r= s(k- t+ 1) -edge-coloured countably infinite complete k-graph can be partitioned into the cores of at most s monochromatic t-tight Berge-paths of different colours. We further describe a construction showing that this result is best possible

    Partitioning Edge-Colored Hypergraphs into Few Monochromatic Tight Cycles

    Get PDF
    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

    Get PDF
    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
    • …
    corecore