Minimum Resolvable Coverings of Kv with Copies of K4 − e

Suppose Kv is the complete undirected graph with v vertices and K4 − e is the graph obtained from a complete graph K4 by removing one edge. Let (K4 − e)-MRC(v) denote a resolvable covering of Kv with copies of K4 − e with the minimum possible number n(v, K4 − e) of parallel classes. It is readily verified that $${n(v, K_4-e) \geq \lceil 2(v-1)/5 \rceil}$$ . In this article, it is proved that there exists a (K4 − e)-MRC(v) with $${\lceil 2(v-1)/5 \rceil}$$ parallel classes if and only if v ≡ 0 (mod 4) with the possible exceptions of v = 108, 172, 228, 292, 296, 308, 412. In addition, the known results on the existence of maximum resolvable (K4 − e)-packings are also improved.