An improvement on Łuczak's connected matchings method

Abstract

A connected matching in a graph G is a matching that is contained in a connected component of G. A well-known method due to Luczak 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.

Ore- and Pósa-type conditions for partitioning $2$-edge-coloured graphs into monochromatic cycles

“…With the help of Szemerédi's regularity lemma [58], the task of finding large cycles in a dense graph G can be relaxed to finding large connected matchings in an appropriately defined reduced graph R. This idea was first used by Komlós, Sárközy and Szemerédi [29] to prove an approximate version of the Pósa-Seymour conjecture and then transferred to monochromatic cycle covers by Luczak [40,41]. Ever since then, the method has become standard practice and fueled numerous advances [2,5,6,15,32,33,37,38], including many of the results mentioned in Section 1. We therefore limit ourselves to stating the necessary definitions and lemmas, mostly following the notation from Lang and Sanhueza-Matamala [34].…”