A new construction for Cohen-Macaulay graphs

Abstract

Let $G$ be a finite simple graph on a vertex set $V(G)=\{x_{11},\dots ,x_{n1}\}$. Also let $m_1,\dots ,m_n\ge 2$ be integers and $G_1,\dots ,G_n$ be connected simple graphs on the vertex sets $V(G_i)=\{x_{i1},\dots ,x_{i{m}_i}\}$. In this paper, we provide necessary and sufficient conditions on $G_1,\dots ,G_n$ for which the graph obtained by attaching $G_i$ to $G$ is unmixed or vertex decomposable. Then we characterize Cohen--Macaulay and sequentially Cohen--Macaulay graphs obtained by attaching the cycle graphs or connected chordal graphs to an arbitrary graphs.

9 pages, 2 figures

References in corpus (2)

• Vertex decomposable graphs and obstructions to shellability
• Algebraic study on Cameron-Walker graphs