Packings and coverings of λKv by k-circuits with one chord

Abstract

Let λK v 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 λK v , denoted by ( v, G, λ)-PD (( v, G, λ)-CD) is a pair (X, B) , where X is the vertex set of K v and 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) 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 C k ( 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⩽ r<⌊ k/2⌋. We give a unified method to construct maximum C k ( r) -packings and minimum C k ( r) -coverings. Especially, for G= C 6 ( r) ( r=1,2), C 7 ( r) ( r=1,2) and C 8 ( r) ( r=1,2,3), we construct the maximum ( v, G, λ)-PD and the minimum ( v, G, λ)-CD.