Large independent sets in regular graphs of large girth

Joseph Lauer, Nicholas Wormald

Research output: Contribution to journalArticleResearchpeer-review

21 Citations (Scopus)

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) ≥ frac(1, 2) (1 - (d - 1)- 2 / (d - 2) - ε{lunate} (g)) n where ε{lunate} (g) → 0 as g → ∞, and we compute explicit bounds on ε{lunate} (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.

Original languageEnglish
Pages (from-to)999-1009
Number of pages11
JournalJournal of Combinatorial Theory, Series B
Volume97
Issue number6
DOIs
Publication statusPublished - Nov 2007
Externally publishedYes

Keywords

  • Differential equation
  • Graph
  • Independence ratio
  • Independent set
  • Large girth

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