Faculté des sciences

Binomial edge ideals of bipartite graphs

Bolognini, Davide ; Macchia, Antonio ; Strazzanti, Francesco

In: European Journal of Combinatorics, 2018, vol. 70, p. 1–25

Binomial edge ideals are a noteworthy class of binomial ideals that can be associated with graphs, generalizing the ideals of 2-minors. For bipartite graphs we prove the converse of Hartshorne’s Connectedness Theorem, according to which if an ideal is Cohen–Macaulay, then its dual graph is connected. This allows us to classify Cohen–Macaulay binomial edge ideals of bipartite graphs, giving... Plus

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    Summary
    Binomial edge ideals are a noteworthy class of binomial ideals that can be associated with graphs, generalizing the ideals of 2-minors. For bipartite graphs we prove the converse of Hartshorne’s Connectedness Theorem, according to which if an ideal is Cohen–Macaulay, then its dual graph is connected. This allows us to classify Cohen–Macaulay binomial edge ideals of bipartite graphs, giving an explicit and recursive construction in graph-theoretical terms. This result represents a binomial analogue of the celebrated characterization of (monomial) edge ideals of bipartite graphs due to Herzog and Hibi (2005). Herzog J., Hibi T. Distributive lattices, bipartite graphs and Alexander duality J. Algebraic Combin., 22 (2005), pp. 289-302