For a graded ideal $I$ of a polynomial ring $S=K[x_1,\ldots,x_n]$ over a field $K$, the Cohen-Macaulayness of the ring $S/I$ may depend on the characteristic of the base field $K$ in general. Even when $I$ is the edge ideal of a graph, this is the case. Nevertheless, for some families of graphs like chordal graphs and K\"onig graphs, the Cohen--Macaulay property of the edge ideal $I(G)$ has combinatorial descriptions which means that it is independent of the base field. In this talk, inspired by the notion of K\"onig graphs, we introduce graded ideals of K\"onig type with respect to a monomial order $<$. It turns out that if $I$ is of K\"onig type, then the Cohen--Macaulay property of $\ini_<(I)$ does not depend on the characteristic of the base field. This happens to be the case also for $I$ itself when $I$ is a monomial ideal or a binomial edge ideal. We consider the ideals of K\"onig type among the edge ideals, binomial edge ideals and the toric ideal of a Hibi ring and use the König property to determine explicitly their canonical module.
This is based on a joint work with Jürgen Herzog and Takayuki Hibi

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