Study On The Adjacent Vertex Distinguishing Total Coloring Of Some Types Of Graphs

Let G be a connected graph of order greater than or equal to 2, k be a positive integer, σ is a mapping from V(G)∪E(G) to {1,2,……,k},if(i) for uv,vw∈E(G),u≠w, σ(uv)≠σ(vw); (ii) for uv∈E(G), σ(u)≠σ(v),σ(u)≠σ(uv),σ(v)≠σ(uv); then σ is called a k- proper total coloring of G.Further.for u∈V(G), σ[u] = {σ(u)}∪{σ(uv)|uv∈E(G),v∈V(G)}. If σ alsosatisfies(iii) for uv∈E(G), σ[u]≠σ[v];then we call σ a k -adjacent-vertex-distinguishing total coloring of G, denoted simply by k-AVDTC, x_at(G)=min{k|G hasa k-adjacent-vertex -distinguishing total coloring } the adjacent vertex distinguishing totalchromatic number.In this thesis, we shall discuss the adjacent vertex distinguishing total coloring of some graphs. In the second chapter we shall determine the adjacent vertex distinguishing total chromatic number of F_r,m,n-graphs: When r = 3,m = l orn-2/w-1 = 0, the adjacent vertex distinguishing total chromatic number of the flower graphs is equal to the sum of its maximum degree and 2; otherwise, the adjacent vertex distinguishing total chromatic number of the flower graphs is equal to the sum of its maximum degree and 1. In the third and fourth chapter, we shall determine the adjacent vertex distinguishing total chromatic number of Halin graphs and 1-Tree graphs: Let G be a Halin graphs of â–³(G)≥5 ,Vv∈I(G) , d(v)≥4 , T = G-E(f₀) and â–³(T) = â–³(G) , then when E(T[V&]) = 0, %at{G) = A(G) + \; when E(t[V,])*0, Zat(G) = A{G) + 2. Let G be a 1-Tree graph of â–³(G)≥4 , then when E(T[V_â–³]) = φ , X_at(G) = â–³(G) + 1 ; when...