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) 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)) 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<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
MSU Digital Commons Citation
Cutler, Jonathan and Pebody, Luke, "Maximal-Clique Partitions and the Roller Coaster Conjecture" (2017). Department of Mathematics Facuty Scholarship and Creative Works. 103.