The Roller-Coaster Conjecture Revisited

Abstract

A graph is well-covered if all its maximal independent sets are of the same cardinality (Plummer, 1970). If G is a well-covered graph, has at least two vertices, and G-v is well-covered for every vertex v, then G is a 1-well-covered graph (Staples, 1975). We call G a λ-quasi-regularizable graph if λ |S| =< |N(S)| for every independent set S of G. The independence polynomial I(G;x) is the generating function of independent sets in a graph G (Gutman & Harary, 1983). The Roller-Coaster Conjecture (Michael & Travis, 2003), saying that for every permutation σ of the set {(α/2),...,α} there exists a well-covered graph G with independence number α such that the coefficients (s_{k}) of I(G;x) are chosen in accordance with σ, has been validated in (Cutler & Pebody, 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 domain of the Roller- Coaster Conjecture can be shortened for well-covered graphs and 1-well-covered graphs.