Geodesic
Decomposition of the Tensor Product of Complete Graphs into Cycles of Lengths 3 and 6
Discussiones Mathematicae. Graph Theory, Tome 41 (2021) no. 1, pp. 249-266

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By a {C_3^α, C_3^β}-decomposition of a graph G, we mean a partition of the edge set of G into α cycles of length 3 and β cycles of length 6. In this paper, necessary and sufficient conditions for the existence of a {C_3^α, C_3^β}-decomposition of (K_m × K_n)(λ), where × denotes the tensor product of graphs and λ is the multiplicity of the edges, is obtained. In fact, we prove that for λ ≥ 1, m, n ≥ 3 and (m, n) ≠ (3, 3), a {C_3^α, C_3^β}-decomposition of (Km × Kn)(λ) exists if and only if λ(m − 1)(n − 1) ≡ 0 (mod 2) and 3α+6β=λ m(m-1)n(n-1)/2.
Keywords: cycle decomposition, tensor product 
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Paulraja, P.; Srimathi, R. Decomposition of the Tensor Product of Complete Graphs into Cycles of Lengths 3 and 6. Discussiones Mathematicae. Graph Theory, Tome 41 (2021) no. 1, pp. 249-266. http://geodesic.mathdoc.fr/item/DMGT_2021_41_1_a15/