Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: 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.

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The categories of this paper are: cond-mat
22
The categories of this paper are: chao-dyn
33
The categories of this paper are: plasm-ph
37
The categories of this paper are: supr-con
19
The categories of this paper are: eess
32
The categories of this paper are: atom-ph
3
The categories of this paper are: physics
5
The categories of this paper are: gr-qc
15
The categories of this paper are: nucl-ex
14
The categories of this paper are: math-ph
28
The categories of this paper are: alg-geom
17
The categories of this paper are: q-fin
30
The categories of this paper are: q-alg
23
The categories of this paper are: solv-int
18
The categories of this paper are: econ
16
The categories of this paper are: stat