We consider the problem of designing fast and compact solutions for representing the connectivity information of triangle meshes. While traditional data structures (Half-Edge, Corner Table) are fast and user-friendly, they tend to be memory-expensive. On the other hand, compression schemes, while meeting information-theory lower bounds, lack the ability to facilitate rapid navigation within the mesh structure. Compact representations provide an advantageous balance for representing large meshes, enabling a judicious compromise between memory consumption and fast implementation of navigational operations. We propose new representations that are sensitive to the regularity of the graph while still having worst case guarantees. For all our data structures we have both an interesting storage cost, typically 2 or 3 r.p.v. (references/vertex) in the case of very regular triangulations and provable upper bounds in the worst case scenario. One of our solution has a worst case cost of 3.33 r.p.v. which is currently the best-known bound improving the previous 4 r.p.v. [Castelli et al. 2018]. In terms of running time, our representations are slightly slower (factor 1.5 to 4) than classical data structures. In our experiments we compare on various meshes runtime and memory performance of our representations with those of the most efficient existing solutions.