Some Results On The Vertex-distinguishing Total Colorings Of The Disjoint Union Of Cycles

Let f be a colouring on vertices and edges of a simple graph G,and f is called a(proper)total colouring of G =(V,E)if the following three conditions are satisfied:(v)No two adjacent vertices receive the same colour;(e)No two adjacent edges receive the same colour;(i)No vertex receives the same colour as any one of its incident edges does.The above-mentioned three conditions are called V-condition,-condition and l-condition,respectively.A general total colouring of a graph G is an assignment of k colours 1,2,…,k to all the vertices and edges of G.Let f be a general total colouring.f is called a V-total colouring if both conditions(e)and(i)are satisfied.Let G be a simple graph.For a total coloring[or a V-total coloring]f of a graph G and any vertex x of G,let Cf(x)or C(x)denote the set,of colors of vertex x and of the edges incident with x,that is C(x)= {f(xu)|xu ? E}(?){f(x)}.We call C(x)the color set of x.If C(u)? C(v)for any two different vertices u and v of V(G),then/is called a vertex-distinguishing total coloring of G[or a vertex-distinguishing V-total coloring of G]and is abbreviated to a VDTC of G[or a VDVTC of G].The minimum number of colors required for a VDTC of G[or a VDVTC of G]is denoted by Xvt(G)[or Xvut(G)]and is called the vertex-distinguishing total chromatic number of G[or vertex-distinguishing V-total chromatic number of G].By the method of distributing prior of the color sets,the vertex-distinguishing total chromatic number of the union mC6,mC7,mC8,mC9 of m disjoint cycles of order 6,7,8,9 will be discussed respectively.And the vertex-distinguishing total chromatic number of mC6,mC7,mC8,mC9 will be obtained in this paper,respectively.