Boosted second moment method in random regular graphs

Abstract

Determining the asymptotic independence ratio of random regular graphs is a key challenge in the area of sparse random graphs. Due to the interpolation method, we have very good upper bounds at our disposal, which are actually known to be sharp for sufficiently large degrees. However, we are still in need of good explicit lower bounds for specific degrees. The classical approach by Frieze and {\L}uczak achieves a lower bound by first applying the second moment method to sparse Erd\H{o}s--R\'enyi graphs, and then cleverly transitioning from that model to regular graphs. They obtain an asymptotic formula (as the degree tends to infinity) but no explicit lower bounds are derived for specific degrees. In contrast, in this paper, we apply the second moment method directly to random regular graphs. This approach has a number of advantages. First, we can numerically compute good explicit lower bounds for any given degree $d$. Moreover, we can even boost this lower bound by arguing that the obtained independent set has a certain spatial Markov property. One can then exploit this property by making local modifications to the independent set, resulting in substantial improvements, and beating the previous best bounds for any $d \geq 10$. Finally, this method gives finer asymptotics as $d \to \infty$ than the original Frieze--{\L}uczak approach. Moreover, these results can be useful even beyond the scope of the independence ratio due to the fact that independent sets with the Markov property may be used to construct other objects in random regular graphs. To demonstrate this, we consider the problem of decomposing random regular graphs into stars.