For n at least 6k^3, every n-vertex graph with minimum degree at least n-k contains every connected n-vertex sparse graph with at most n(1+1/(24k-12)) edges and alpha' at least k-1.
Hall, On representatives of subsets, J
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Minimum degree and sparse connected spanning subgraphs
For n at least 6k^3, every n-vertex graph with minimum degree at least n-k contains every connected n-vertex sparse graph with at most n(1+1/(24k-12)) edges and alpha' at least k-1.