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- W1456426994 abstract "A graph G is well-covered if every maximal independent set has the same cardinality q . Let i k ( G ) denote the number of independent sets of cardinality k in G . Brown, Dilcher, and Nowakowski conjectured that the independence sequence ( i 0 ( G ) , i 1 ( G ) , … , i q ( G ) ) was unimodal for any well-covered graph G with independence number q . Michael and Traves disproved this conjecture. Instead they posited the so-called “Roller Coaster” Conjecture: that the terms i ⌈ q 2 ⌉ ( G ) , i ⌈ q 2 ⌉ + 1 ( G ) , … , i q ( G ) could be in any specified order for some well-covered graph G with independence number q . Michael and Traves proved the conjecture for q < 8 and Matchett extended this to q < 12 . In this paper, we prove the Roller Coaster Conjecture using a construction of graphs with a property related to that of having a maximal-clique partition. In particular, we show, for all pairs of integers 0 ≤ k < q and positive integers m , that there is a well-covered graph G with independence number q for which every independent set of size k + 1 is contained in a unique maximal independent set, but each independent set of size k is contained in at least m distinct maximal independent sets." @default.
- W1456426994 created "2016-06-24" @default.
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- W1456426994 date "2017-01-01" @default.
- W1456426994 modified "2023-10-17" @default.
- W1456426994 title "Maximal-clique partitions and the Roller Coaster Conjecture" @default.
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- W1456426994 doi "https://doi.org/10.1016/j.jcta.2016.06.019" @default.
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