Alternate proof of the Caro-Wei theorem for lower bounding the independence number
An algorithmic proof is given by Murphy [1, Section 1], where he attributes the algorithm to Erdös [2].

As a side remark, I don't know what the original proof of Caro or Wei is, but I believe the proof you mention is due to Alon and Spencer [3].



[1] Murphy, Owen. "Lower bounds on the stability number of graphs computed in terms of degrees." Discrete Mathematics 90.2 (1991): 207-211.

[2] Erdös, Paul. "On the graph theorem of Turán." Mat. Lapok 21. (1970): 249-251.

[3] Alon, Noga, and Joel H. Spencer. The Probabilistic Method. John Wiley &amp; Sons, 2004.