Vertex coverings by monochromatic paths and cycles

Abstract

We survey some results on covering the vertices of 2-colored complete graphs by two paths or by two cycles of different color. We show the role of these results in determining path Ramsey numbers and in algorithms for finding long monochromatic paths or cycles in 2-colored complete graphs.A graph is 2-colored if each edge is colored either red or blue. We consider results on covering the vertices of 2-colored complete (undirected, directed, or bipartite) graphs by monochromatic paths or cycles. Algorithmic proofs of these results are given, and these results are applied to the calculation of generalized Ramsey numbers.A 2-colored path or cycle is called simple if it is monochromatic or it is the union of a red and a blue path. The following simple result was mentioned in a footnote in [ 2]. Proof (Algorithm 1). Assume that n ~ 2.-We define simple paths Pi,

Monochromatic paths in random tournaments

“…Indeed, the directed size Ramsey number of P n is Θ(n 2 ), as follows from results by Raynaud [12] (see also [8] for a simple proof) and Reimer [13].…”

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

“…(Gyárfás [19], 1983) The vertex set of any 2-colored complete graph can be covered by a red and a blue cycle that intersect in at most one vertex.…”

Recurrence of Space-Time Events

“…The classical Ramsey Theorem has since been generalized in many ways; most of these generalizations are concerned with finding monochromatic substructures in various colored structures [20]- [23]. Monochromatic paths and cycles have been investigated by many authors, for example see [24]- [26].…”