On the binomial edge ideals of proper interval graphs
  We prove several cases of the Betti number conjecture for the binomial edge ideal $J_G$ of a proper interval graph $G$ (also known as closed graph). Namely, we show that this conjecture is true for the linear strand of $J_G$, and true in general for any proper interval graph $G$ such that the regularity of $S/J_G$ equals two. 
Herolistra Baskoroputro
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