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Independence Sequences of Well-Covered Graphs: Non-Unimodality and the Roller-Coaster Conjecture

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Abstract.

A graph G is well-covered provided each maximal independent set of vertices has the same cardinality. The term s k of the independence sequence (s 0,s 1,…,s α) equals the number of independent k-sets of vertices of G. We investigate constraints on the linear orderings of the terms of the independence sequence of well-covered graphs. In particular, we provide a counterexample to the recent unimodality conjecture of Brown, Dilcher, and Nowakowski. We formulate the Roller-Coaster Conjecture to describe the possible linear orderings of terms of the independence sequence.

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Correspondence to William N. Traves.

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We are grateful to Jason Brown for a helpful discussion in December of 1999 and to Richard Stanley for valuable feedback in August of 2000. The work of the second author was partially supported by the Naval Academy Research Council and ONR grant N0001400WR20041

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Michael, T., Traves, W. Independence Sequences of Well-Covered Graphs: Non-Unimodality and the Roller-Coaster Conjecture. Graphs and Combinatorics 19, 403–411 (2003). https://doi.org/10.1007/s00373-002-0515-7

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  • DOI: https://doi.org/10.1007/s00373-002-0515-7

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