Bounds on the independence number of a graph in terms of order, size and maximum degree

• We prove several best-possible lower bounds in terms of the order and the average degree for the independence number of graphs which are connected and/or satisfy some odd girth condition. Our main result is the extension of a lower bound for the independence number of triangle-free graphs of maximum degree at most 3 due to Heckman and Thomas [A New Proof of the Independence Ratio of Triangle-Free Cubic Graphs, Discrete Math. 233 (2001), 233-237] to arbitrary triangle-free graphs. For connected triangle-free graphs of order n and size m, our result implies the existence of an independent set of order at least (4n−m− 1)/7.