Study On Some Types Of Chromatic Sum And Strengths Of Graphs

A best coloring of G is a mapping f from the vertex set V to a color set C such that any two neighborly vertice have different images and (?)f(v) obtains the minimumvalue.We call the minimum value of (?)f(v) is minimal vertex chromatic sum of G,and is denoted by∑(G),namely∑(G)=min (?)f(v).If f is a best coloring of G,wecall the minimal color number that f needs is the vertex-strength of G ,and is denoted by s(G).For general graphs,s(G)≥x(G).In the second chapter,we shall study the∑(G) and s(G) of paths,cycles,stars,wheels,fans,complete bipartite graphs and hypo-complete bipartite graphs,then we shall give the s(G) of Cartesian product graphs,hexagonal systems,F_r,m,n-graphs,C_m?F_n graphs,and one type of k-th power graphs.A best edge coloring of G is a mapping f from the edge set E to a color set Csuch that any two neighborly edges have different images and (?)f(e) obtains the minimumvalue.We call the minimum value of (?)f(e) is the minimal edge chromatic sum of G ,and is denoted by∑'(G),namely∑'(G)=min (?)f(e)-If f is a best edge coloring of G ,wecall the minimal color number that f needs is the edge-strength of G,and is denoted bys'(G).At the present time,there is few conclusions of the study about∑'(G) and s'(G).In thefourth chapter,we shall study the∑'(G) and s'(G) of the hypo-complete bipartite graphs.At the base of the conception of total coloring,we shall extend two above conceptions and define the concepts of the best coloring and the strength of graph G.Let f be a mapping from {V,E} to {1,2,…,k}.If(1) for (?)u,v∈V,uv∈E,u≠v,f(u)≠f(v); (2) for (?)uv,uw∈E,v≠w,f(uv)≠f(uw);(3) for (?)u,v∈V,uv∈E,u≠v,f(u)≠f(uv),f(v)≠f(uv),then f is called a k - proper total coloring of G .If f is a total coloring of G ,we let∑"(G)=min (?)(f(v)+f(e)),and call∑"(G) is the minimal total coloring sum.We callthe minimal color number that f needs is the strength of G,and is denoted by s_T(G).Using the techniques of exchanging colors and enumeration,we shall determine∑"(G) and s_T(G) of several kinds of graphs in the third chapter.