On characterizing the critical graphs for matching Ramsey numbers

Given simple graphs $H_{1},H_{2},\ldots,H_{c}$, the Ramsey number $r(H_{1},H_{2},\ldots,H_{c})$ 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 $i\in\{1,2,\ldots,c\}$. The critical graphs for $r(H_1,H_2,\ldots,H_c)$ are edge-colored complete graphs on $r(H_1,H_2,\ldots,H_c)-1$ vertices with $c$ colors which contain no subgraphs in color $i$ isomorphic to $H_{i}$ for any $i\in \{1,2,\ldots,c\}$. For $n_1\geq n_2\geq \ldots\geq n_c\geq 1$, Cockayne and Lorimer (The Ramsey number for stripes, {\it J.\ Austral.\ Math.\ Soc.} \textbf{19} (1975), 252--256.) showed that $r(n_{1}K_{2},n_{2}K_{2},\ldots,n_{c}K_{2})=n_{1}+1+ \sum\limits_{i=1}^c(n_{i}-1)$, in which $n_{i}K_{2}$ is a matching of size $n_{i}$. Using the Gallai-Edmonds Theorem, we characterized all the critical graphs for $r(n_{1}K_{2},n_{2}K_{2},\ldots,n_{c}K_{2})$, implying a new proof for this Ramsey number.