Extremal graphs and classification of planar graphs by MC-numbers

Abstract

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 that mc(G)In this paper we depict all graphs with mc(G)We also prove that mc(G) ≤ m − n + 4 if G is a planar graph, and classify all planar graphs by their monochromatic connectivity numbers.