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An edge-coloring of a connected graph $G$ is called a {\em monochromaticconnection coloring} (MC-coloring for short) if any two vertices of $G$ areconnected by a monochromatic path in $G$. For a connected graph $G$, the {\emmonochromatic connection number} (MC-number for short) of $G$, denoted by$mc(G)$, is the maximum number of colors that ensure $G$ has a monochromaticconnection coloring by using this number of colors. This concept was introducedby 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 allplanar graphs by their monochromatic connectivity numbers.

Extremal graphs and classification of planar graphs by MC-numbers