Random d-regular graphs a.a.s. have k-star decompositions with s or s+1 stars per vertex for all k < d/2 - 1, with prescribed star centers when k is sufficiently small.
Nowhere-zero 3-flows and modulo k-orientations.Journal of Combinatorial Theory, Series B, 103(5):587–598, 2013
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Star decompositions via orientations
Random d-regular graphs a.a.s. have k-star decompositions with s or s+1 stars per vertex for all k < d/2 - 1, with prescribed star centers when k is sufficiently small.