The reference given in [the Wikipedia article on linkless embedding][1] 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.



  [1]: https://en.wikipedia.org/wiki/Linkless_embedding