The Roller-Coaster Conjecture Revisited

A graph is well-covered if all its maximal independent sets are of the same cardinality (Plummer in J Comb Theory 8:91–98, 1970). If G is a well-covered graph with at least two vertices, and G- v is well-covered for every vertex v, then G is a 1-well-covered graph (Staples in On some subclasses of well-covered graphs. Ph.D. Thesis, Vanderbilt University, 1975). The graph G is λ-quasi-regularizable if λ> 0 and λ· | S| ≤ | N(S) | for every independent set S of G. It is known that every well-covered graph without isolated vertices is 1-quasi-regularizable (Berge in Ann Discret Math 12:31–44, 1982). The independence polynomialI(G;x)=∑k=0αskxk is the generating function of independent sets in a graph G (Gutman and Harary in Util Math 24:97–106, 1983), where α is the independence number of G. The Roller-Coaster Conjecture (Michael and Traves in Graphs Comb 19:403–411, 2003), saying that for every permutation σ of the set {…,α} there exists a well-covered graph G with the independence number α such that the coefficients (s_k) of I(G; x) satisfy (Formula Presented.) has been validated in Cutler and Pebody (J Comb Theory A 145:25–35, 2017). In this paper we show that independence polynomials of λ-quasi-regularizable graphs are partially unimodal. More precisely, the coefficients of an upper part of I(G; x) are in non-increasing order. Based on this finding, we prove that the unconstrained part of the independence sequence is: (Formula Presented.) for well-covered graphs, and (Formula Presented.) for 1-well-covered graphs, where α stands for the independence number, and n is the cardinality of the vertex set.