Decomposing random regular graphs into stars
Michelle Delcourt, Catherine Greenhill, Mikhail Isaev, Bernard Lidický, Luke Postle · European Journal of Combinatorics · 2025
We study $k$-star decompositions, that is, partitions of the edge set into disjoint stars with $k$ edges, in the uniformly random $d$-regular graph model $\mathcal{G}_{n,d}$. Using the small subgraph conditioning method, we prove an existence result for such decompositions for all $d,k$ such that $d/2 d/2$. For smaller values of $k$, the connection between $k$-star decompositions and $β$-orientations allows us to apply results of Thomassen (2012) and Lovász, Thomassen, Wu and Zhang (2013). We prove that random $d$-regular graphs satisfy their assumptions with high probability, thus establishing a.a.s. existence of $k$-star decompositions (i) when $2k^2+k\leq d$, and (ii) when $k$ is odd and $k < d/2$.