Map Design, Plus A Hanging Side Of The K <sub> 2, S </ Sub>

A complete multigraph of order υ and index A. denoted by XK_?. is an undirected graph with υ vertices, where any two distinct vertices .x and y are joined by λ edges {x.y}. Let G be a finite simple graph. A graph design G-GD_λ(υ) of XK(?) is a pair (X.β) where X is the vertex set of K_υ and β is a collection of subgraphs of K_υ. called blocks, .such that each block is isomophic to G and any two distinct vertices in K_υ are joined in exact A blocks of B.In this paper, the graph design of K_2,s with a pendent edge, including two kinds of graph. P_s and Q_s, is researched. From [3]. we have known the existence results of a P_s-GD(v) and a Q_s-GD(υ), where υ = 0,1 (mod 2.s + 1). In this paper, the futher results are given for the case 2s + I = 3q and gcd(3,q) = 1.