Maximal-clique partitions and the Roller Coaster Conjecture

TL;DR

This paper proves the Roller Coaster Conjecture, demonstrating that the order of certain independence sequence terms in well-covered graphs can be arbitrarily arranged, using a novel graph construction related to maximal-clique partitions.

Contribution

It introduces a new graph construction method to prove the Roller Coaster Conjecture for all independence numbers, extending previous partial results.

Findings

The conjecture holds for all independence numbers q.

Constructed graphs have unique maximal independent sets for certain sizes.

Graphs can have multiple independent sets of smaller sizes contained in larger ones.

Abstract

A graph $G$ is {\em well-covered} if every maximal independent set has the same cardinality $q$. Let $i_k(G)$ denote the number of independent sets of cardinality $k$ in $G$. Brown, Dilcher, and Nowakowski conjectured that the independence sequence $(i_0(G), i_1(G), \ldots, i_q(G))$ was unimodal for any well-ordered graph $G$ with independence number $q$. Michael and Traves disproved this conjecture. Instead they posited the so-called ``Roller Coaster" Conjecture: that the terms \[ i_{\left\lceil\frac{q}2\right\rceil}(G), i_{\left\lceil\frac{q}2\right\rceil+1}(G), \ldots, i_q(G) \] could be in any specified order for some well-covered graph $G$ with independence number $q$. Michael and Traves proved the conjecture for $q<8$ and Matchett extended this to $q<12$. In this paper, we prove the Roller Coaster Conjecture using a construction of graphs with a property related to that of having a maximal-clique partition. In particular, we show, for all pairs of integers $1\le k<q$ and positive integers $m$, that there is a well-covered graph $G$ with independence number $q$ for which every independent set of size $k+1$ is contained in a unique maximal independent set, but each independent set of size $k$ is contained in at least $m$ distinct independent sets.