Universality of highly imbalanced generated graphs

Prove that the graph generated by a rooted tree whose descendant-subtree imbalance ratio is allowed to be as large as K = Ω(ε^{-3}) contains every n-vertex spanning tree, without adding edges from each vertex to the nearest-right cousin and all of that cousin’s descendants.

Background

The paper studies sparse universal graphs for the family of all n-vertex trees. Its construction generates a graph from a rooted tree by adding edges to descendants, left siblings and their descendants, nearest-left cousins and their descendants, and, in the r-generated variant, selected descendants of ancestors or nearest-left cousins.

The authors observe that allowing a large imbalance between the descendant-subtree sizes of a vertex and its nearest-left cousin can substantially reduce the number of edges. Specifically, they state that when this ratio is as large as K = Ω(ε{-3}), a tree T can be constructed with e(G_Tr) ≤ (5/2 + ε)n log₂ n. However, the embedding argument needed to prove universality becomes difficult because some required edges between separated descendant subtrees may be absent. The authors obtain universality only after adding edges toward nearest-right cousins, at the cost of a weaker bound, and leave the universality of the sparser construction unresolved.

References

If we allow the ratio between \nu(l(u)) and \nu(u) to be as large as K=\Omega(\varepsilon{-3}), then we can construct a tree T with e(Gr_T)\leq (\frac{5}{2}+\varepsilon) n \log_2 n. However, we were not able to prove that such a graph contains all possible spanning trees, as the imbalance imposes additional difficulty to embed an arbitrary tree T' into Gr_T.

— On the size of universal graphs for spanning trees  (2508.19032 - Kim et al., 26 Aug 2025) in Section 2, subsection “Proof overview”