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

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.