Packings and Coverings for Ten Graphs With Seven Points, Seven Edges and an Even-Cycle

Abstract

Let \(\lambda K_v\) be the complete multigraph with \(v\) vertices, where any two distinct vertices \(x\) and \(y\) are joined by \(\lambda\) edges \(\{x,y\}\). Let \(G\) be a finite simple graph. A \(G\)-packing design (\(G\)-covering design) of \(K_v\), denoted by \((v, G, \lambda)\)-PD \(((v, G,\lambda)\)-CD), is a pair \((X, \mathcal{B})\), where \(X\) is the vertex set of \(K_v\), and \(\mathcal{B}\) is a collection of subgraphs of \(K_v\), called blocks, such that each block is isomorphic to \(G\) and any two distinct vertices in \(K_v\) are joined in at most (at least) \(\lambda\) blocks of \(\mathcal{B}\). A packing (covering) design is said to be maximum (minimum) if no other such packing (covering) design has more (fewer) blocks. In this paper, we have completely determined the packing number and covering number for the graphs with seven points, seven edges and an even cycle.