(d,1)-total Labelling On Graphs And Coloring Game

A (d, 1)-total labelling of graph G is an assignment of integers to vertices and edges of graph G such that any two adjacent vertices of G receive distinct integers, any two adjacent edges of G receive distinct integers, and a vertex and its incident edges receive labels that differ in absolute value by at least d. The span of a (d, 1)-total labelling is the maximum difference between any two labels. The (d, 1)-total number is the least span among all the (d, 1)-total labellings of G. In the second chapter, we obtain the (d, 1)-total numbers of stars, trees and equal complete tripartite graphs.In the third chapter, it is studied that the (d, 1)-total labelling of The Sierpinski graphs S(n,k), Sierpinski gasket graphs Sn. graphs S+(n,k) and S++(n,k). Moreover, it is determinedλdT (S(n,k)),λdT(Sn),λdT(S+(n,k)) andλdT(S++(n,k)) for any d≥k≥3 and n≥2.If every vertex is labelled with the label belongs to l∈[0,Δ-d-1] in a (d, 1)-total labelling, the labelling is called d-good labelling. We give a total labelling method in the forth chapter to prove that for any graph G withΔ(G)≥2d+2, there is a d-good labelling of Gin [0.2Δ+d-2]. It's better than the known upper bound 2Δ+d-1.Let f be a graph function which assigns to H a non-negative integer such that f(H)≤v(H). In the last chapter f is defined as f(K2)= 2, f(P8)= 3,f(C2n)= 3 and f(H)= 0 otherwise. It is proved that the f-acyclic game chromatic number of an unicycle graph with just an even cycle is 9 at most.