Classification of commutative zero-divisor semigroup graphs (Q613528)

Let \(S\) denote a commutative semigroup with \(0.\) The associated zero-divisor graph \(\Gamma(S)\) is the graph whose vertices form a one-to-one correspondence with the non-zero elements of \(S\) and two vertices are joined by an edge if the product of the corresponding elements in \(S\) is zero. Given a graph \(G\), does there exist a zero-divisor semigroup \(S\) such that \(G = \Gamma(S)\)? In this case \(G\) is called admissible. The results of the paper are the following: {\parindent=7mm \begin{itemize}\item[(1)]Given a graph \(G\) with connected components \(G_{\alpha}.\) Then \(G\) is admissible if and only if each \(G_{\alpha}\) is, provided that none of the \(G_{\alpha}\) is a singleton. \item[(2)]Every graph (not necessarily connected) is the induced subgraph of a connected admissible graph. \item[(3)]The authors introduce the notion of depth as a new invariant. They reformulate the four admissibility conditions of \textit{F. DeMeyer} and \textit{L. DeMeyer} [J. Algebra 283, No. 1, 190--198 (2005; Zbl 1077.20069)], in terms of the depth and give several other admissibilty conditions in terms of the depth. \item[(4)]They define the notion of graph separability. It is used for a coloring criterion to determine a graph to be non-admissible. \item[(5)]They settle in the affirmative the question: Can be any regular surface (or 2-manifold) be triangulated in such a way that it is the graph of some commutative zero-divisor semigroup? \end{itemize}}