Minimum Coverings of the Complete Graph with 5-cycles

Let \( K_v \) be the complete graph on \( v \) vertices, and \( C_5 \) be a cycle of length five. A simple minimum \( (v, C_5, 1) \)-covering is a pair \( (V, C) \) where \( V = V(K_v) \) and \( C \) is a family of edge-disjoint 5-cycles of minimum cardinality which partition \( E(K_v) \cup E \), for some \( E \subset E(K_v) \). The collection of edges \( E \) is called the excess. In this paper, we determine the necessary and sufficient conditions for the existence of a simple minimum \( (v, C_5, 1) \)-covering. More precisely, for each \( v \geq 6 \), we prove that there is a simple minimum \( (v, C_5, 1) \)-covering having all possible excesses.