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Let Meta( H > T, λ) denote the set of all integers v such that there exists a ( H > T)- GM ( v). In this paper, the set Meta( H > T, λ) will be completely determined for the following 21 pairs ( H, T) = ( H, P), ( H, 2 P), ( H, P) and ( H, P), where $${P_2 \subset H_1 \subseteq K_4, H_2 \in \{K_4, C_4, K_3+e, P_4\} = {\mathcal {H}}, H_3 \in {\mathcal {H}} \cup \{K_3, K_{1,3} \}}$$ and $${H_4 \in \{K_4, C_4\}}$$ . [ABSTRACT FROM AUTHOR]

The maximum packing C- max PD( v) and minimum covering C- minCD( v) of K with 8-cycles are studied, all structures with the nonisomorphic leave (excess) are presented. In Li et al. (Graphs Combin 25:735-752, ), C- max PD( v) and C- minCD( v) have been determined for odd v. In this paper, we introduce the enumeration of nonisomorphic $${(v,\frac{v}{2}+s)}$$-graphs ( s = 4, 6), give complete solution of the maximum packing and minimum covering designs of K with 8-cycles for any even v with all possible leaves (excesses). [ABSTRACT FROM AUTHOR]

Let λKv be the complete multigraph with v vertices, where any two distinct vertices x and y are joined by λ edges {x,y}. Let G be a finite simple graph. A G-packing design (G-covering design) of λKv, denoted by (v,G,λ)-PD ((v,G,λ)-CD) is a pair (X,B), where X is the vertex set of Kv and B is a collection of subgraphs of Kv, called blocks, such that each block is isomorphic to G and any two distinct vertices in Kv are joined in at most (at least) λ blocks of B. A packing (covering) design is said to be maximum (minimum) if no other such packing (covering) design has more (fewer) blocks. In this paper, the discussed graphs are Ck(r), i.e., one circle of length k with one chord, where r is the number of vertices between the ends of the chord, 1&les;r<⌊k/2⌋. We give a unified method to construct maximum Ck(r)-packings and minimum Ck(r)-coverings. Especially, for G=C6(r)(r=1,2), C7(r)(r=1,2) and C8(r)(r=1,2,3), we construct the maximum (v,G,λ)-PD and the minimum (v,G,λ)-CD. [Copyright &y& Elsevier]