Decomposition of hypercube graphs into paths and cycles having k edges

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For an even graph G and positive integer p, q, and k, the pair (p, q) is an admissible pair if ( p + q ) k = | E ( G ) | . If a graph G admits a decomposition into p copies of P k + 1 , the path of length k, and q copies of Ck , the cycle of length k, for every admissible pair (p, q), then G has a { P k + 1 , C k } { p , q } -decomposition. In this paper, we give necessary and sufficient conditions for the existence of a { P k + 1 , C k } { p , q } -decomposition of n-dimensional hypercube graphs Qn when n is even, k ≥ 4 , and n ≡ 0 ( mod k ) .