Giancarlo Rinaldo, Francesco Romeo: $2$-Dimensional vertex decomposable circulant graphs, 301-320

Abstract:

Let $G$ be the circulant graph $C_n(S)$ with $S\subseteq\{ 1,\ldots,\left \lfloor\frac{n}{2}\right \rfloor\}$ and let $\Delta$ be its independence complex. We describe the well-covered circulant graphs with 2-dimensional $\Delta$, and construct an infinite family of vertex-decomposable circulant graphs within this family. Moreover, we show that if $C_n(S)$ has a 2-dimensional vertex decomposable $\Delta$, then it has a level Stanley-Reisner ring.

Key Words: Circulant graphs, Cohen-Macaulay, vertex decomposability.

2010 Mathematics Subject Classification: Primary 13F55; Secondary 13H10.