  In this paper, we give a new criterion for the Cohen-Macaulayness of vertex
splittable ideals, a family of monomial ideals recently introduced by Moradi
and Khosh-Ahang. Our result relies on a Betti splitting of the ideal and
provides an inductive way of checking the Cohen-Macaulay property. As a result,
we obtain characterizations for Gorenstein, level and pseudo-Gorenstein vertex
splittable ideals. Furthermore, we provide new and simpler combinatorial proofs
of known Cohen-Macaulay criteria for several families of monomial ideals, such
as (vector-spread) strongly stable ideals and (componentwise) polymatroidals.
Finally, we characterize the family of bi-Cohen-Macaulay graphs by the novel
criterion for the Cohen-Macaulayness of vertex splittable ideals.
