Extremal graphs and classification of planar graphs by MC-numbers
Yanhong Gao, Ping Li, Xueliang Li · Discussiones Mathematicae Graph Theory · 2021
A path in an edge-colored graph is called monochromatic if all the edges in the path have the same color.An edge-coloring of a connected graph G is called a 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 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 thatif G is a planar graph, and classify all planar graphs by their monochromatic connection numbers.