On Well-Covered, Vertex Decomposable and Cohen-Macaulay Graphs

Iván D. Castrillón, Roberto Cruz, Enrique Reyes

Abstract


Let $G=(V,E)$ be a graph. If $G$ is a König graph or if $G$ is a graph without 3-cycles and 5-cycles, 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 König type $e_{1},\dots,e_{g}$ without 4-cycles with two $e_i$'s. Furthermore, we study vertex decomposable and shellable (non-pure) properties in graphs without 3-cycles and 5-cycles. Finally, we give some properties and relations between critical, extendable and shedding vertices.


Keywords


Cohen-Macaulay rings, Shellable simplicial complex, Well-covered graphs, Vertex decomposable, König graphs

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