An exact upper bound on the size of minimal clique covers

Indeterminate strings have received considerable attention in the recent past; see for example [1] and [3]. This attention is due to their applicability in bioinformatics, and to the natural correspondence with undirected graphs. One aspect of this correspondence is the fact that the minimal alphabet size of indeterminates representing any given undirected graph corresponds to the size of the minimal clique cover of this graph. This paper solves a related problem proposed in [3]: compute Θn(m), which is the size of the largest possible minimal clique cover (i.e., an exact upper bound), and hence alphabet size of the corresponding indeterminate, of any graph on n vertices and m edges.