The Monochromatic Connection Number Of Graphs

The edge coloring of graphs is one of the hot issue in graphs theory.The connec-tivity of edge-colored graphs has been one of the most active topics in the past ten years.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 concepts were introduced by Caro&Yuster in 2011.In recent decades,the monochromatic connection theory has been studied extensivelyIn this master thesis,we study the monochromatic connection number of graphs A straightforward lower bound for mc(G)is m(G)-n(G)+2[4].Y.Caro&Y.Yuster also given some upper bounds for mc(G),one of them is mc(G)?m(G)-m(G)+x(G)The followings are main contentsIn the first chapter,the basic concepts and terminology are introduced.The research background and current status of monochromatic connection number are also elaborated in detail,and finally the main results are described brieflyIn the second chapter,we study the monochromatic connection number of graph G with ?(G)=3 and we solve completely the monochromatic connectivity of 3-chromatic graphsIn the third chapter,we study the monochromatic connection number of r-partite graphs,and we give two new upper bounds for mc(G),which are characterized by the structures of extremal coloringsIn the last chapter,we consider the characterization of the upper bounds of the monochromatic connection number,and we characterize all connected graphs G achieving the upper bounds mc(G)=m-n+?(G),mc(G)=m-n+?(G)-1.