We show that for any ε > 0 and ∆ ∈ N, there exists α > 0 such that for sufficiently large n, every n-vertex graph G satisfying that δ(G) ≥ εn and e(X, Y ) > 0 for every pair of disjoint vertex sets X, Y ⊆ V (G) of size αn contains all spanning trees with maximum degree at most ∆. This strengthens a result of Böttcher et al.
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