Independent dominating sets in graphs of girth five Ararat Harutyunyan ∗ Paul Horn † Jacques Verstraete ‡ March 9, 2009 Abstract One of the first results in probabilistic combinatorics is that every n-vertex graph of minimum degree at least d contains a dominating set of size at most n(1+log(d+1)) d+1 . In this paper, we show that every n-vertex d-regular graph of girth at least five contains an independent dominating set of size at most n(log d+c) d for some constant c. Apart from the value of the constant c, this result is best possible in view of known results on independent dominating sets in random d-regular graphs. In addition, the condition that the graph has girth at least five is necessary since the smallest independent dominating set in an n-vertex graph whose components are all isomorphic to K d,d is exactly n 2 . 1 Introduction An early result using the probabilistic method is that every n-vertex graph of minimum degree at least d contains a dominating set of size at most n(1+log(d+1)) d+1 . This result is due independently to Lov´ asz [6], Payan [9] and Arnautov [2]. In this paper, we prove the following: Theorem 1. Let n, d be positive integers such that n> 10d 2 log 2 d. Then there exists an absolute constant c such that any n-vertex d-regular graph of girth at least five contains an independent dominating set of size at most n(log d+c) d . We make no attempt to optimize the constant c here. This theorem is best possible in view of random d-regular graphs: asymptotically almost surely, all independent dominating sets in * Department of Mathematics and Statistics, McGill University † Department of Mathematics, University of California at San Diego. Email: phorn@math.ucsd.edu ‡ Department of Mathematics, University of California at San Diego, 9500 Gilman Drive, La Jolla, California 92093-0112, USA; Research supported by an Alfred P. Sloan Research Fellowship and an NSF Grant DMS 0800704. Email: jverstra@math.ucsd.edu 1