On well-covered, vertex decomposable and Cohen-Macaulay graphs

TL;DR

This paper explores the equivalence of several algebraic and combinatorial properties in specific classes of graphs, such as well-covered, vertex decomposable, and Cohen-Macaulay graphs, providing characterizations and relations among these properties.

Contribution

It establishes new equivalences and characterizations for well-covered, vertex decomposable, and Cohen-Macaulay graphs, especially those without certain cycles, and studies properties of unicyclic graphs.

Findings

Equivalence of shellability, Cohen-Macaulayness, and vertex decomposability in certain graphs

Characterization of well-covered graphs without 3-, 5-, and 7-cycles

Properties of unicyclic graphs with various algebraic and combinatorial conditions

Abstract

Let $G=(V,E)$ be a graph. If $G$ is a K\"onig graph or $G$ is a graph without 3-cycles and 5-cycle, we prove that the following conditions are equivalent: $\Delta_{G}$ is pure shellable, $R/I_{\Delta}$ is Cohen-Macaulay, $G$ is unmixed vertex decomposable graph and $G$ is well-covered with a perfect matching of K\"onig type $e_{1},...,e_{g}$ without square with two $e_i$'s. We characterize well-covered graphs without 3-cycles, 5-cycles and 7-cycles. Also, we study when graphs without 3-cycles and 5-cycles are vertex decomposable or shellable. Furthermore, we give some properties and relations between critical, extendables and shedding vertices. Finally, we characterize unicyclic graphs with each one of the following properties: unmixed, vertex decomposable, shellable and Cohen-Macaulay.