Decomposition of the tensor product of complete graphs into cycles of lengths 3 and 6

By a {C3α, C3β} \{ C_3^\alpha ,\,C_3^\beta \} -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 {C3α, C3β} 3\alpha + 6\beta = {{\lambda m(m - 1)n(n - 1)} \over 2} -decomposition of (Km × Kn)(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 {C3α, C3β} -decomposition of (Km × Kn)(λ) exists if and only if λ(m − 1)(n − 1) ≡ 0 (mod 2) and 3α+6β=λm(m-1)n(n-1)2 .