Cohen-Macaulayness of a class of graphs versus the class of their complements. (English) Zbl 1467.13010
Summary: Let \(n \geq 2\) be an integer. The graph \(G(n)\) is obtained by letting all the elements of \(\{0, \ldots, n - 1 \}\) to be the vertices and defining distinct vertices \(x\) and \(y\) to be adjacent if and only if \(\gcd(x + y, n) = 1\). In this paper, well-coveredness, Cohen-Macaulayness, vertex-decomposability and Gorensteinness of these graphs and their complements are characterized. These characterizations provide large classes of Cohen-Macaulay and non Cohen-Macaulay graphs.
MSC:
| 13A70 | General commutative ring theory and combinatorics (zero-divisor graphs, annihilating-ideal graphs, etc.) |
| 05E40 | Combinatorial aspects of commutative algebra |
| 13H10 | Special types (Cohen-Macaulay, Gorenstein, Buchsbaum, etc.) |
| 13F55 | Commutative rings defined by monomial ideals; Stanley-Reisner face rings; simplicial complexes |
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