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Tensor models generalize the matrix-model approach to 2-dimensional quantumgravity to higher dimensions. Some models allowing a $1/N$ expansion have beenexplored, most of them generating branched-polymer geometries. Recently,enhancements yielding an additional 2d quantum-gravity (planar) phase and anintermediate regime of proliferating baby-universes have been found. It remainsan open issue to find models escaping these lower dimensionality universalityclasses.  Here we analyse the dominant regime and critical behaviour of a range of newmodels which are candidates for such effective geometries, in particularinteractions based on the utility graph $K_{3,3}$. We find that, upon properenhancement, the two-phase structure of a branched-polymer and a 2d gravityregime is the common case in $U(N)$-invariant rank $D=4$ tensor models of smallorders. Not only the well known so-called necklace interactions but also$K_{3,3}$-type interactions turn out as the source for the planar regime. Wegive a systematic account of the enhancement scaling, the counting ofleading-order diagrams and the multi-critical behaviour of a wide range ofinteractions, in particular for all order-6 interactions of rank 3 and 4. Thesefindings support the claim of universality of such mixtures of branched-polymerand planar diagrams at criticality. In particular, this hints at the necessityto consider new ingredients, or interactions of higher order and rank, in orderto obtain higher dimensional continuum geometry from tensor models.

Multi-critical behaviour of 4-dimensional tensor models up to order 6