Abstract
For an even graph G and non-negative integers r and s, the pair \((r,\,s)\) is an admissible pair if \(4r+8s=|E(G)|.\) If G admits a decomposition into r copies of \(C_4,\) the cycle of length four, and s copies of \(C_8,\) the cycle of length eight, for every admissible pair \((r,\,s),\) then G has a \(\{C_4^r,\,C_8^s\}\)-decomposition. In this paper, a necessary and sufficient condition is obtained for the existence of a \(\{C_4^r,\,C_8^s\}\)-decomposition of the complete k-partite graph \(K_{a_1,\,a_2,\,\dots ,\,a_k},\) where \(k\ge 3.\) Further, a characterization is obtained for the graph \(K_m\times K_n,\) where \(\times\) denotes the tensor product of graphs, to admit a \(\{C_4^r,\,C_8^s\}\)-decomposition.
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Ganesamurthy, S., Paulraja, P. Decompositions of Some Classes of Dense Graphs into Cycles of Lengths 4 and 8. Graphs and Combinatorics 37, 1291–1310 (2021). https://doi.org/10.1007/s00373-021-02317-6
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DOI: https://doi.org/10.1007/s00373-021-02317-6