Monochromatic connected matchings in 2-edge-colored multipartite graphs

József Balogh, University of Illinois Urbana-Champaign
Alexandr Kostochka, University of Illinois Urbana-Champaign
Mikhail Lavrov, Kennesaw State University
Xujun Liu, Xi'an Jiaotong-Liverpool University


A matching (Formula presented.) in a graph (Formula presented.) is connected if all the edges of (Formula presented.) are in the same component of (Formula presented.). Following Łuczak, there have been many results using the existence of large connected matchings in cluster graphs with respect to regular partitions of large graphs to show the existence of long paths and other structures in these graphs. We prove exact Ramsey-type bounds on the sizes of monochromatic connected matchings in 2-edge-colored multipartite graphs. In addition, we prove a stability theorem for such matchings.