Δ-Optimum Forbidden Subgraphs and Exclusive Sum Labellings of Graphs

The notions of sum labelling and sum number of graphs were introduced by F. Harary [1] in 1990. A mapping f is called a sum labelling of a graph G (V, E) if it is an injection from V to a set of positive integers such that uv is an element of E if and only if there exists a vertex w is an element of V such that f(w) = f (x) f (y). In this case, w is called a working vertex. If f is a sum labelling of G boolean OR rK(1) for some nonnegative integer r and G contains no working vertex, f is defined as an exclusive sum labelling of the graph C by M. Miller et al. in paper [2]. The least possible number r of such isolated vertices is called the exclusive sum number of G, denoted by epsilon(G). If epsilon(G) = Delta(G), the labelling is called Delta-optimum exclusive sum labelling and the graph is said to be is Delta-optimum summable, where Delta = Delta(G) denotes the maximum degree of vertices in G. By using the notion of Delta-optimum forbidden subgraph of graph the exclusive sum numbers of crown C-n circle dot K-1 and (C-n circle dot K-1) are given in this paper. Some Delta-optimum forbidden subgraphs of trees are studied and we prove that for any integer Delta >= 3 there exist trees not Delta-optimum summable, and a nontrivial upper bound of the exclusive sum numbers of trees is also given in this paper.