Graded ideals of K\"onig type

Inspired by the notion of K"onig graphs we introduce graded ideals of K"onig type with respect to a monomial order $<$. It is shown that if $I$ is of K"onig type, then the Cohen–Macaulay property of ${\text{ini}}_{<}(I)$ does not depend on the characteristic of the base field. This happens to be the case also for $I$ itself when $I$ is a binomial edge ideal. Attached to an ideal of K"onig type is a sequence of linear forms, whose elements are variables or differences of variables. This sequence is a system of parameters for ${\text{ini}}_{<}(I)$, and is a potential system of parameters for $I$ itself. We study in detail the ideals of K"onig type among the edge ideals, binomial edge ideals and the toric ideal of a Hibi ring and use the K"onig property to determine explicitly their canonical module.