The Roller-Coaster Conjecture Revisited

TL;DR

This paper investigates the properties of independence polynomials in well-covered and related graphs, showing partial unimodality and refining the scope of the Roller-Coaster Conjecture.

Contribution

It demonstrates partial unimodality of independence polynomials in {}-quasi-regularizable graphs and refines the domain of the Roller-Coaster Conjecture for specific graph classes.

Findings

Independence polynomials of {}-quasi-regularizable graphs are partially unimodal.

The upper part of the independence polynomial coefficients is non-increasing.

The domain of the Roller-Coaster Conjecture is shortened for well-covered and 1-well-covered graphs.

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 {\lambda}-quasi-regularizable graph if {\lambda} |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 {\sigma} of the set {({\alpha}/2),...,{\alpha}} there exists a well-covered graph G with independence number {\alpha} such that the coefficients (s_{k}) of I(G;x) are chosen in accordance with {\sigma}, has been validated in (Cutler & Pebody, 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 domain of the Roller- Coaster Conjecture can be shortened for well-covered graphs and 1-well-covered graphs.