Vertex-distinguishing ?-total Colorings And ?-total Colorings Of Several Joins Of Graphs

A general total coloring of G is an assignment of several colors to all vertices and edges of G such that any two adjacent vertices and any two adjacent edges of G are assigned different colors,that is called 1-total coloring of a graph G.For an general total coloring of a graph G if any two adjacent edges of G are assigned different colors,that is called VI-total coloring of a graph G.Let f be an I(VI)-total coloring of a graph G and any vertex u of G,let Cf(u)or C(u)denote the set of colors of vertex u and of the edges incident with u,that is C(u)= {f(uv)|uv ? E} U {f(u)}.We called C(u)is the color set of u.If C(u)? C(v)fur any two different vertices u and v of G,then f is called a vertex-distinguishing I-total coloring(or vertex distinguishing VI-total coloring)of G,or VDITC(or VDVITC)of G for short.The minimum number of colors required in a VDITC(or VDVITC)is the vertex-distinguishing I-total chromatic number(or VI-total chromatic number),is denoted byxvti(G)(orXvtvi(G)).In this paper,we discuss problems of vertex-distinguishing I-total colorings and vertex-distinguishing VI-total colorings of the join of cycle and path by the method of constructing concrete coloring.Meanwhile,vertex-distinguishing I-total chromatic numbers and vertex-distinguishing Vl-total chromatic numbers of Pm ? Pn,Pm ? Cn,Cm ? Cn,Cm ? Wn,Cm ? Fn,Pm?Wn,Pm? Fn are determined.The results in this paper illustrated that the VDITC conjecture and VDVITC conjecture are valid for Pm ? Pn,Pm ? Cn,Cm ? Cn,Cm ? Wn,Cm?Fn,Pm?Wn,Pm?Fn.