Cyclically Interval Total Colorings Of Some Cycle Related Graphs

A total coloring of a graph G is a function ?:E{G)? V{G)?N such that no adjacent vertices,edges,and no incident vertices and edges obtain the same color.A k-interval is a set of k consecutive integers.A cyclically interval total t-coloring of a graph G is a total coloring a of G with colors 1,2,···,t such that at least one vertex or edge of G is colored by i.i?1,2….t,and for any v ? V(G),the set S[?,v]={?(v)} ? {?(e)|e is incident to v} is a(dc(v)+1)-interval,or ?1,2,...,t}\S[?,v]is a(t-dG(v)-1)-interval,where dc(v)is the degree of the vertex v in G.A middle graph M(G)of a graph G is the graph whose vertex set is V(G)U E(G)and in which two vertices are adjacent whenever either they are adjacent edges of G or one is a vertex of G and other is an edge incident with it.A total graph T(G)of a graph G is the graph whose vertex set is V(G)UE(G)and in which two vertices are adjacent whenever they are either adjacent or incident in G.A shadow graph S(G)of a graph G is constructed by taking two copies of G,say G" and G".Join each vertex u' in G" to the neighbors of the corresponding vertex u" in G".The one point union Cn(k)of k-copies of cycle Cn is the graph obtained by taking v as a common vertex such that any two distinct cycles C'n and Cn" are edge disjoint and do not have any vertex in common except v.In this paper,we study the cyclically interval total colorings of M((Cn),T((Cn),S(Cn)and Cn(k),respectively,for any integers n? 3 and k?2,show that all these graphs are cyclically interval total colorable,get the exact values of w?c(G),and provide the lower bounds of W?c(G),where G is M(Cn),T(Cn),S(Cn)or Cn(k}.