[Paper Review] Shellable and Cohen-Macaulay complete t-partite graphs

Seyyede Masoome Seyyedi, Farhad Rahmati+1 more|arXiv (Cornell University)|Apr 10, 2012

Commutative Algebra and Its Applications3 references3 citations

TL;DR

This paper characterizes shellability, vertex decomposability, and Cohen-Macaulayness in complete t-partite graphs via combinatorial and algebraic conditions. It proves that a complete t-partite graph is shellable (and vertex decomposable) if and only if exactly one color class has more than one vertex and all others are singletons, while it is Cohen-Macaulay if and only if all color classes have exactly one vertex—equivalent to being a complete graph K_t.

ABSTRACT

Let G be a simple undirected graph. We find the number of maximal independent sets in complete t-partite graphs. We will show that vertex decomposability and shellability are equivalent in this graphs. Also, we obtain an equivalent condition for Cohen-Macaulay in complete t-partite graphs.

Motivation & Objective

• To determine the number of maximal independent sets in complete t-partite graphs.
• To establish equivalent conditions for shellability, vertex decomposability, and Cohen-Macaulayness in complete t-partite graphs.
• To characterize when a complete t-partite graph is Cohen-Macaulay via its color class structure.
• To clarify the relationship between shellability, vertex decomposability, and Cohen-Macaulayness in this graph class.

Proposed method

• Uses the simplicial complex Δ_G associated with the graph G, where faces are independent sets.
• Applies the definition of shellability via ordering of facets (maximal independent sets) with specific intersection properties.
• Employs vertex decomposability via recursive removal of vertices satisfying link and deletion conditions.
• Uses the equivalence between chordal graphs and vertex decomposability in the context of complete t-partite graphs.
• Applies Herzog-Hibi-Zheng theorem linking Cohen-Macaulayness and unmixedness in chordal graphs.
• Analyzes minimal vertex covers and their cardinalities to determine unmixedness and Cohen-Macaulayness.

Experimental results

Research questions

• RQ1 When is a complete t-partite graph shellable?
• RQ2 What is the relationship between shellability and vertex decomposability in complete t-partite graphs?
• RQ3 Under what conditions is a complete t-partite graph Cohen-Macaulay?
• RQ4 How does the structure of color classes affect the Cohen-Macaulay property in complete t-partite graphs?
• RQ5 When do shellability and Cohen-Macaulayness coincide in complete t-partite graphs?

Key findings

• The number of maximal independent sets in a complete t-partite graph is exactly t.
• A complete t-partite graph is shellable if and only if exactly one color class has more than one vertex and all others are singletons.
• Vertex decomposability in complete t-partite graphs is equivalent to the same condition on color classes as for shellability.
• A complete t-partite graph is Cohen-Macaulay if and only if all color classes have exactly one vertex, i.e., G ≅ K_t.
• Cohen-Macaulayness and shellability coincide in complete t-partite graphs precisely when G is a complete graph K_t.