Independence, odd girth, and average degree

Abstract

We prove several tight lower bounds in terms of the order and the average degree for the independence number of graphs that 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 three due to Heckman and Thomas [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. © 2010 Wiley Periodicals, Inc. J Graph Theory 67:96-111, 2011 © 2011 Wiley Periodicals, Inc.