The reference given in the Wikipedia article on linkless embedding for the $4n-10$ bound on the number of edges in a linkless embeddable graph is Mader, W. (1968), "Homomorphiesätze für Graphen", Mathematische Annalen 178 (2): 154–168, doi:10.1007/BF01350657. Apparently Mader proves this bound more generally for $K_6$-minor-free graphs. As the Wikipedia article also states, the example of apex graphs shows that this is tight.
David Eppstein
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