A search is proposed for dibaryonic systems with non-zero charm and beauty quantum numbers which are more suitable for unambiguous identification. An estimate of the production cross sections for the dibaryonic systems H c , H b and H bs is presented.Searches for exotic states like the glueball and multiquark hadronic states have not yielded any conclusive evidence so far. Part of the difficulty lies in the fact that the states that have been searched for have the same quantum numbers as the conventional mesonic and baryonic resonances and therefore are difficult to distinguish from the same. In this paper we make a case for looking for specific dibaryonic states which are more suitable for unambiguous identification though they may be more difficult to produce or detect. Table 1 displays a few selected dibaryonic states [1] in the SU(3), SU(4), and SU(5) flavor sectors with their masses and other quantum numbers and the hadronic particle channels to which they can communicate.The masses were calculated using the MIT bag model. Consider the dibaryons -°iH 5/2 and ~\H 3/2 . We use the notation J s Hj for a dibaryon with spin, strangeness and isospin quantum numbers J, s, and /. From the table it can be seen that in all the possible decay modes only strangeness -1 dibaryon H s/2 has a decay mode with a final state whose threshold energy is less than the mass of the dibaryon. But this mode will also be suppressed since parity conservation demands that it be at least a p-wave transition, and further it violates flavor symmetry. Thus it is reasonable to expect that these states are much longer lived and more likely to decay by weak interactions, thereby becoming more amenable to unambiguous identification.* Presented at a Workshop in honor of E. C. G. Sudarshan's contributions to Theoretical Physics, held at the University of Texas in Austin, September 15-17, 1991.Reprint requests to Dr. G. Bhamathi. The production of the dibaryons discussed above in pp and Kd collisions is, however, much more difficult that those that have already been studied, since they require the transfer of either a large amount of strangeness or isospin or both. Further, the threshold energies for these processes are very large and the cross sections are likely to be rather small. In the SU (4) and SU(5) sectors we find that the lightest dibaryon states are: From Table 1 it can be seen that all the lowest threshold final states have energies much greater than the masses of the corresponding dibaryons, which are therefore expected to decay only by weak interactions. We now proceed to estimate the production cross sections for these states.