A connected matching is a matching contained in a connected component. A well-known method due to Łuczak reduces problems about monochromatic paths and cycles in complete graphs to problems about monochromatic matchings in almost complete graphs. We show that these can be further reduced to problems about monochromatic connected matchings in complete graphs.
I will describe Łuczak's reduction, introduce the new reduction, and mention potential applications of the improved method.
- Combinatorial Theory Seminar