Large independent sets in regular graphs of large girth

Authors:
Joseph Lauer;Nicholas Wormald
Affiliations:
Department of Combinatorics and Optimization, University of Waterloo, Waterloo ON, Canada;Department of Combinatorics and Optimization, University of Waterloo, Waterloo ON, Canada
Venue:
Journal of Combinatorial Theory Series B
Year:
2007

Citing 1
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Abstract

Let G be a d-regular graph with girth g, and let @a be the independence number of G. We show that @a(G)=12(1-(d-1)^-^2^/^(^d^-^2^)-@e(g))n where @e(g)-0 as g-~, and we compute explicit bounds on @e(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 of 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.