Constraining MC-numbers by the connectivity of complement graphs

Abstract

An edge-coloring of a graph G is called a monochromatic connection coloring (or MC-coloring for short) if any two vertices of G are joined by a monochromatic path (a path whose edges are colored with the same color). The monochromatic connection number (or MC-number for short) of G, denoted by mc(G), is the maximum number of colors used in an MC-coloring of G. In this paper we main explore the relationships between mc(G) and the connectivity of its complement graph G‾. We give a method for computing mc(G) if G‾ is disconnected and give the sharp upper bounds of mc(G) when G‾ is k-connected, where 1≤k≤3. If G is a graph with G‾k-connected and k≥4, then mc(G)=m−n+2 (this result has been proved by Caro and Yuster) and we further talk about the characteristics of MC-colorings of G. In addition, we prove that any graph G with mc(G)≥m−n+t contains a Kt.