From logical gates synthesis to chromatic bicritical clutters Original Research Article

The synthesis of logical gates by means of basic gates leads very naturally to the concept of clones of boolean functions and the corresponding lattice of clones largely studied by E. Post; a deep investigation into the clones of “positive functions” leads to the problem of characterizing their minimal functions; the canonical bijection between positive functions and Sperner hypergraphs (also called clutters) allows one to define (associated with these minimal functions) chromatic bicritical clutters which are nothing else than clutters H having the property that, for every edge e, the clutter whose edges are those of H disjoint from e, has a chromatic number two less than that of H. The bicritical clutters which are graphs are the source of a conjecture from Lovász (1966) which is still open: complete graphs are the only bicritical graphs! This survey is an attempt to grasp the (combinatorial and gate-theoretical) properties of these clutters, many of which inherit from some well-known properties of self-transversal clutters (which are exactly the non-trivial bicritical clutters of chromatic number three). New results are given for these clutters (mainly in corresponding gates synthesis). Even if older results have been proved and appeared in previous papers (from 1965 to 1983), we have choosen to recall all these proofs in order to make the paper self-contained.