Large independent sets in regular graphs of large girth

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Abstract

Let G be a d-regular graph with girth g, and let α be the independence number of G. We show that α(G)12(1(d1)2/(d2)ϵ(g))n where ϵ(g)0 as g, and we compute explicit bounds on ϵ(g) for small g. For large g this improves previous results for all d7. 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.

Keywords

Graph
Independent set
Large girth
Independence ratio
Differential equation

Cited by (0)

1

Supported by NSERC via a USRA. Current address: Department of Mathematics, Yale University, New Haven, CT, USA.

2

Supported by the Canada Research Chairs Program and NSERC.