Decomposition of Complete Tripartite Graphs into Short Cycles

For a graph G and for non-negative integers p, q and r, the triplet (p, q, r) is said to be an admissible triplet, if 3 p + 4 q + 6 r = | E (G) |. If G admits a decomposition into p cycles of length 3, q cycles of length 4, and r cycles of length 6 for every admissible triplet (p, q, r), then we say that G has a C p 3, C q 4, C r 6 -decomposition. In this paper, the necessary conditions for the existence of C p 3, C q 4, C r 6 -decomposition of K ℓ, m, n (ℓ ≤ m ≤ n) are proved to be sufficient. This affirmatively answers the problem raised in Decomposing complete tripartite graphs into cycles of lengths 3 and 4, Discrete Math. 197/198 (1999), 123-135. As a corollary, we deduce the main results of Decomposing complete tripartite graphs into cycles of lengths 3 and 4, Discrete Math. , 197/198, 123-135 (1999) and Decompositions of complete tripartite graphs into cycles of lengths 3 and 6, Austral. J. Combin. , 73 (1), 220-241 (2019).