Finding a maximum independent set in a graph is well known to be an NP-complete problem. Here an $O (n ^ 2) $- time algorithm. That finds an independent set of order at least $(6N - M) / 13 $in a triangle-free graph with n vertices and m edges is, presented. A tight lower bound on independence in 4-regular triangle-free graphs is $4N / 13 $, so the bound is sharp for this class.
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