Maximal-clique partitions and the Roller Coaster Conjecture

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.