Some Cohen–Macaulay and Unmixed Binomial Edge Ideals

Abstract

We study unmixed and Cohen-Macaulay properties of the binomial edge ideal of some classes of graphs. We compute the depth of the binomial edge ideal of a generalized block graph. We also characterize all generalized block graphs whose binomial edge ideals are Cohen–Macaulay and unmixed. So that we generalize the results of Ene, Herzog, and Hibi on block graphs. Moreover, we study unmixedness and Cohen–Macaulayness of the binomial edge ideal of some graph products such as the join and corona of two graphs with respect to the original graphs.

Key Words:

• Binomial edge ideal
• Cohen-Macaulay
• Corona of graphs
• Generalized block graph
• Join of graphs
• Unmixed

2010 Mathematics Subject Classification:

• 13C05
• 05E40

Notes

Communicated by S. Goto.