Unmixed and Cohen--Macaulay weighted oriented K\"onig graphs

29 Sep 2019  ·  Pitones Yuriko, Reyes Enrique, Villarreal Rafael H. ·

Let $D$ be a weighted oriented graph, whose underlying graph is $G$, and let $I(D)$ be its edge ideal. If $G$ has no $3$-, $5$-, or $7$-cycles, or $G$ is K\"{o}nig, we characterize when $I(D)$ is unmixed... If $G$ has no $3$- or $5$-cycles, or $G$ is K\"onig, we characterize when $I(D)$ is Cohen--Macaulay. We prove that $I(D)$ is unmixed if and only if $I(D)$ is Cohen--Macaulay when $G$ has girth greater than $7$ or $G$ is K\"onig and has no $4$-cycles. read more

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Commutative Algebra Combinatorics