On complementary path decompositions of the complete multigraph

We give a complete solution to the existence problem for complementary P3-decompositions of the complete multigraph, where P 3 denotes the path of length 3. A complementary decomposition 2>.K v (P 3 , P 3) is an edge decomposition of the complete multigraph >'Kv into P 3 '8 with the property that upon taking the complement of each path one obtains a second decomposition of >'K3 into P3 '8 (where the complement of the path abcd is the path bdac). The following result was proven by Granville, Moisiadis and Rees in [1] (and, with a few small exceptions, also follows from the techniques in [3]): Theorem 1. There exists a complementary decomposition 2Kv ~ (P 3 , P 3) if and only if v == 1 (mod 3). In this paper, we give a complete solution to the existence problem for complementary P 3-decompositions of the complete multigraph 2)'K v o Note that if D is such a decomposition then the set {(a, b, c, d) : abcd E D} is an edge decomposition of 2>.K v into K4 's, that is, a (v, 4, 2>.)-BIBD. The following result was proven by Ranani in [2]: Lemma 2. If there exists a complementary decomposition 2>.K v ~ (P 3 , P 3), then >.(v-1) == 0 (mod 3) and >.v(v-1) == 0 (mod 6). As a consequence of the remarks above and the results in Ranani [2], we need consider only the case >. = 3.