Let l be an odd integer.It was conjectured that every l-regular graph containing a perfect matching can be decomposed into paths of length l.For the case l=5,Favaron et al.(2010)verified the conjecture for graphs with no cycle of length 4,and Botler et al.(2015)verified it for triangle-free graphs.In this paper,we prove that every 5-regular graph with a perfect matching can be decomposed into paths of length 5,provided that 3-cycles and 4-cycles in the graph have no edge in common.
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