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$$P_4$$-decompositions of regular graphs. (English) Zbl 0927.05067
It is shown that every simple $$r$$-regular graph $$G$$ admits a balanced $$P_4$$-decomposition if $$r \equiv 0\pmod 3$$ and $$G$$ has no cut-edge when $$r$$ is odd. It is also shown that a connected 4-regular graph $$G$$ admits a $$P_4$$-decomposition if and only if $$| E(G)| \equiv 0\pmod 3$$ by characterizing graphs of maximum degree 4 that admit a triangle-free Eulerian tour.

##### MSC:
 05C70 Edge subsets with special properties (factorization, matching, partitioning, covering and packing, etc.) 05C38 Paths and cycles 05C45 Eulerian and Hamiltonian graphs 05C75 Structural characterization of families of graphs
##### Keywords:
regular graphs; path; decomposition; Euler tour
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##### References:
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