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.