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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
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