Large independent sets in regular graphs of large girth

Let G be a d-regular graph with girth g, and let α be the independence number of G. We show that α(G) ≥ 1 2 1 − (d − 1)−2/(d−2) − (g) n where (g) → 0 as g → ∞, and we compute explicit bounds on (g) for small g. For large g this improves previous results for all d ≥ 7. The method is by analysis of a simple greedy algorithm which was motivated by the differential equation method used to bound independent set sizes in random regular graphs. We use a “nibble”-type approach but require none of the sophistication of the usual nibble method arguments, relying only upon a difference equation for the expected values of certain random variables. The difference equation is approximated by a differential equation (though we do not use the “differential equation method”).

• Type: Journal
• Year: 2007
• Where: JCT
• Authors: Joseph Lauer, Nicholas C. Wormald
• Keywords: Difference Equation, Differential Equation Method, JCT 2007, Simple Greedy Algorithm