Geodesic
Decompositions of Complete Bipartite Graphs and Complete Graphs Into Paths, Stars, and Cycles with Four Edges Each
Discussiones Mathematicae. Graph Theory, Tome 41 (2021) no. 2, pp. 451-468

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Let G be either a complete graph of odd order or a complete bipartite graph in which each vertex partition has an even number of vertices. In this paper, we determine the set of triples (p, q, r), with p, q, r gt; 0, for which there exists a decomposition of G into p paths, q stars, and r cycles, each of which has 4 edges.
Keywords: complete graph, complete bipartite graph, path, star, cycle, decomposition 
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     author = {Shyu, Tay-Woei},
     title = {Decompositions of {Complete} {Bipartite} {Graphs} and {Complete} {Graphs} {Into} {Paths,} {Stars,} and {Cycles} with {Four} {Edges} {Each}},
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Shyu, Tay-Woei. Decompositions of Complete Bipartite Graphs and Complete Graphs Into Paths, Stars, and Cycles with Four Edges Each. Discussiones Mathematicae. Graph Theory, Tome 41 (2021) no. 2, pp. 451-468. http://geodesic.mathdoc.fr/item/DMGT_2021_41_2_a7/