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

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. 1 Covers by paths and cycles In this survey r-coloring always means edge-coloring with r colors (traditionally red and blue when r = 2). Some part of the material is already discussed in the 2008 survey of Kano and Li [32]. Roughly fifty years ago, in my very first paper [17] the following statement appeared as a footnote. Proposition 1. (Gerencsér, Gyárfás [17],1967) The vertex set of any 2-edge-colored complete graph Kn can be partitioned into a red and a blue path. For the sake of the rigorous, empty paths and one-vertex paths are accepted as a monochromatic path of any color in Proposition 1. To prove it, suppose that R and B are vertex disjoint red and blue paths with endpoints r, b in a complete graph with a red-blue edge coloring and v / ∈ V (R) ∪ V (B). Then either rv extends R to a red path or bv extends B to a blue path or through the edge rb either R∪ rb∪ bv, B−{b} ∗Research supported in part by the OTKA Grant No. K104343.