On characterizing the critical graphs for matching Ramsey numbers

Given simple graphs H-1, H-2, . . . , H-c, the Ramsey number r(H-1, H-2, . . . , H-c) is the smallest positive integer n such that every edge-colored K-n with c colors contains a subgraph in color i isomorphic to H-i for some i is an element of{1, 2, . . . , c}. The critical graphs for r(H-1, H-2, . . . , H-c) are edge-colored complete graphs on r(HH1, H-2, . . . , H-c) - 1 vertices with c colors which contain no subgraphs in color i isomorphic to Hi for any i. {1, 2, . . . , c}. For n(1) >= n(2) >= . . . >= n(c) >= 1, Cockayne and Lorimer (1975) showed that r(n(1)K(2), n(2)K(2), . . . , n(c)K(2)) = n(1) + 1 + Sigma(c)(i=1)(n(i) - 1), in which n(i)K(2) is a matching of size ni. Using the Gallai-Edmonds Theorem, we characterized all the critical graphs for r(n(1)K(2), n(2)K(2), . . . , n(c)K(2)), implying a new proof for this Ramsey number. (C) 2020 Elsevier B.V. All rights reserved.