Decompositions of λKv into k-circuits with one chord

Let λKv be the complete multigraph with v vertices, where any two distinct vertices x and y are joined by λ edges {x, y}. LetGbeafinite simple graph. A G-design of λKv, denoted by (v, G, λ)-GD, isapair (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 exactly λ blocks of B. In this paper, the graphs discussed are C (r) k, 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⌋. We give a unified method to construct C(r) 2 k-designs. In particular, for G = C (r