Maximal-clique partitions and the Roller Coaster Conjecture

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 (i₀(G),i₁(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 termsi_⌈q2⌉(G),i_⌈q2⌉+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.