Geodesic
On well-covered, vertex decomposable and Cohen-Macaulay graphs
Iván D. Castrillón1 ; Roberto Cruz2 ; Enrique Reyes1
1Departamento de Matemáticas CINVESTAV-IPN
2Instituto de Matemáticas, Universidad de Antioquia
The electronic journal of combinatorics, Tome 23 (2016) no. 2
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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.
DOI : 10.37236/5874
Classification : 13F55, 05E40, 05E45, 05C75
Mots-clés : Cohen-Macaulay, well-covered, unmixed, vertex decomposable, shellable, König, girth, unicyclic

Iván D. Castrillón  1   ; Roberto Cruz  2   ; Enrique Reyes  1

1 Departamento de Matemáticas CINVESTAV-IPN
2 Instituto de Matemáticas, Universidad de Antioquia 
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Iván D. Castrillón; Roberto Cruz; Enrique Reyes. On well-covered, vertex decomposable and Cohen-Macaulay graphs. The electronic journal of combinatorics, Tome 23 (2016) no. 2. doi: 10.37236/5874

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