Some Octagonal Diagram Fill Coverage

A complete multigraph of order v with indexλ, denoted byλK_v, is an undirected graph withv vertices, where any two distinct vertices x and y are joined byλedges {x, y}. Let G be a finitesimple graph. A graph design G-GD_λ(v) (graph packing G-PD_λ(v), graph covering G-CD_λ(v) ofλK_v is a pair (X, B) where X is the vertex set of K_v and B is a collection of subgraphs of K_v, calledblocks, such that each block is isomorphic to G and any two distinct vertices in K_v are joined in exact(at most, at least)λblocks of B. A graph packing (covering) is said to be maximum (minimum) ifno other such graph packing (covering) with the same order has more (fewer) blocks.In this paper, the maximum graph packing and the minimum graph covering of two graphsD₁, D₂ with six vertices and eight edges and the 8-cycle C₈ are researched. By an unified method, wesolve the problems of the maximum graph packing and the minimum graph covering for all possiblev and anyλ(for graphs D₁ and D₂) orλ= 1 (for graph C₈) by recurrence constructions.In addition, the graph design of complete bipartite graph K_2,s with a pendent edge, includingtwo kinds of graph, P_s and Q_s, is researched. From [4], the existence of P_s-GD(v) and Q_s-GD(v)for v = 0,1 (mod 2s + 1) has been known. In this paper, the further results are given for thecase 2s + 1 = 5q and gcd(5,q) =1, discuss the existence of P_s-GD(v) and Q_s-GD(v) for v≡20i + 6,30t + 10 (mod 50t + 15) when q = 10t + 3 and v≡40t + 36,10t + 10 (mod 50t + 45)when q =10t + 9.