Monochromatic connected matchings in 2‐edge‐colored multipartite graphs

Abstract

A matching M $M$ in a graph G $G$ is connected if all the edges of M $M$ are in the same component of G $G$ . 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.

An improvement on Łuczak's connected matchings method

“…Let 𝑠, 𝑘 0 ⩽ 𝑘 be integers, let 𝛽 ⩾ 𝛼 > 0, and let 𝜀 > 0 be sufficiently small. Set 𝛿 = 2 ⋅ ( 56𝑠 [1,15]). One can think of the graphs in the above statements as being multicolour.…”

“…We will find submatchings 𝑀 0 ⊇ 𝑀 1 ⊇ … ⊇ 𝑀 𝑘 such that |𝑀 𝓁 | ⩾ 1 4 ⋅ |𝑀 𝓁−1 | and there are no 𝓁-coloured edges between 𝑋 𝓁 ∶= 𝑋 0 ∩ 𝑉(𝑀 𝓁 ) and 𝑌 𝓁 ∶= 𝑌 0 ∩ 𝑉(𝑀 𝓁 ). Assuming such submatchings are found, then…”

“…Remark It is sometimes convenient to allow edges to have multiple colours (see, for example, [1, 15]). One can think of the graphs in the above statements as being multicolour.…”