\(P_4\)-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.


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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