Extremal graphs with maximum monochromatic connectivity

Abstract

An edge-coloring of a connected graph is called a monochromatic connection coloring (MC-coloring, for short) if there is a monochromatic path joining each two vertices of the graph. For a connected graph G, the monochromatic connection number of G, denoted by mc(G), is defined to be the maximum number of colors allowed in an MC-coloring of G. This concept was introduced by Caro and Yuster (2011). They proved that |E(G)|−|V(G)|+2≤mc(G)≤|E(G)|−|V(G)|+χ(G) for any connected graph G. Some graphs achieving the lower bound have been determined. In this paper, we first give a new upper bound for mc(G), which is characterized by the structures of extremal colorings. Using this new bound, we then characterize all connected graphs G achieving the upper bound mc(G)=|E(G)|−|V(G)|+χ(G).