Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Extremal graphs and classification of planar graphs by MC-numbers   An edge-coloring of a connected graph $G$ is called a {\em monochromatic
connection coloring} (MC-coloring for short) if any two vertices of $G$ are
connected by a monochromatic path in $G$. For a connected graph $G$, the {\em
monochromatic connection number} (MC-number for short) of $G$, denoted by
$mc(G)$, is the maximum number of colors that ensure $G$ has a monochromatic
connection coloring by using this number of colors. This concept was introduced
by Caro and Yuster in 2011. They proved that $mc(G)\leq m-n+k$ if $G$ is not a
$k$-connected graph. In this paper we depict all graphs with $mc(G)=m-n+k+1$
and $mc(G)= m-n+k$ if $G$ is a $k$-connected but not $(k+1)$-connected graph.
We also prove that $mc(G)\leq m-n+4$ if $G$ is a planar graph, and classify all
planar graphs by their monochromatic connectivity numbers.

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The categories of this paper are: hep-ph
29
The categories of this paper are: funct-an
17
The categories of this paper are: q-fin
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The categories of this paper are: stat
12
The categories of this paper are: nucl-th
8
The categories of this paper are: hep-ex
27
The categories of this paper are: comp-gas
23
The categories of this paper are: solv-int
25
The categories of this paper are: patt-sol
33
The categories of this paper are: plasm-ph
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The categories of this paper are: alg-geom
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The categories of this paper are: physics
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The categories of this paper are: quant-ph
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The categories of this paper are: econ
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The categories of this paper are: supr-con
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The categories of this paper are: eess