The Roller-Coaster Conjecture Revisited

Abstract

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 $$\lambda $$ -quasi-regularizable if $$\lambda >0$$ and $$\lambda \cdot \vert S\vert \le \vert N( S) \vert $$ 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 polynomial $$I(G;x)= {\sum _{k=0}^{\alpha }} s_{k}x^{k}$$ is the generating function of independent sets in a graph G (Gutman and Harary in Util Math 24:97–106, 1983), where $$\alpha $$ 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 $$\sigma $$ of the set $$\{\lceil \frac{\alpha }{2}\rceil ,\ldots ,\alpha \}$$ there exists a well-covered graph G with the independence number $$\alpha $$ such that the coefficients $$( s_{k}) $$ of I(G; x) satisfy $$\begin{aligned} s_{\sigma \left( \left\lceil \frac{\alpha }{2}\right\rceil \right) }<s_{\sigma \left( \left\lceil \frac{\alpha }{2}\right\rceil +1\right) }<\cdots <s_{\sigma (\alpha )}, \end{aligned}$$ has been validated in Cutler and Pebody (J Comb Theory A 145:25–35, 2017). In this paper we show that independence polynomials of $$\lambda $$ -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: $$\begin{aligned} \left( s_{\left\lceil \frac{\alpha }{2}\right\rceil },s_{\left\lceil \frac{\alpha }{2}\right\rceil +1},\ldots ,s_{\min \left\{ \alpha ,\left\lceil \frac{n-1}{3}\right\rceil \right\} }\right) \end{aligned}$$ for well-covered graphs, and $$\begin{aligned} \left( s_{\left\lceil \frac{2\alpha }{3}\right\rceil },s_{\left\lceil \frac{2\alpha }{3}\right\rceil +1},\ldots ,s_{\min \left\{ \alpha ,\left\lceil \frac{n-1}{3}\right\rceil \right\} }\right) \end{aligned}$$ for 1-well-covered graphs, where $$\alpha $$ stands for the independence number, and n is the cardinality of the vertex set.