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 \emph{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 \emph{Decomposing complete tripartite graphs into cycles of lengths 3 and 4 , Discrete Math., 197/198, 123-135 (1999)} and \emph{Decompositions of complete tripartite graphs into cycles of lengths 3 and 6 , Austral. J. Combin., 73(1), 220-241 (2019)}.