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 ik(G)ik(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),…,iq(G))(i0(G),i1(G),…,iq(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),…,iq(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<8q<8 and Matchett extended this to q<12q<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