On Decomposing Complete 4-partite Graphs into 3-Cycles and 4-Cycles

In this paper, we first give the necessary and sufficient conditions for a complete 4-partite graphs to decomposed into triangles. Then, by the similar technique, we also prove that if a complete 4-partite graph Kn1n2n3n4 can be decomposed into 3-cycles and the size is equal to 3α + 4β , then the complete 4-partite graph can be decomposed into α 3-cycles and β 4-cycles.