Independent sets and cuts in large-girth regular graphs

We present a local algorithm producing an independent set of expected size 0.44533n on large-girth 3-regular graphs and 0.40407n on large-girth 4-regular graphs. We also construct a cut (or bisection or bipartite subgraph) with 1.34105n edges on large-girth 3regular graphs. These decrease the gaps between the best known upper and lower bounds from 0.0178 to 0.01, from 0.0242 to 0.0123 and from 0.0724 to 0.0616, respectively. We are using local algorithms, therefore, the method also provides upper bounds for the fractional coloring numbers of 1/0.44533 ≈ 2.24554 and 1/0.40407 ≈ 2.4748 and fractional edge coloring number 1.5/1.34105 ≈ 1.1185. Our algorithms are applications of the technique introduced by Hoppen and Wormald. [7]