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Indeterminate strings have received considerable attention in the recentpast; see for example Christodoulakis et al 2015 and Helling et al 2017. Thisattention is due to their applicability in bioinformatics, and to the naturalcorrespondence with undirected graphs. One aspect of this correspondence is thefact that the minimal alphabet size of indeterminates representing any givenundirected graph corresponds to the size of the minimal clique cover of thisgraph. This paper solves a related problem proposed in Helling et al 2017:compute $\Theta_n(m)$, which is the size of the largest possible minimal cliquecover (i.e., an exact upper bound), and hence alphabet size of thecorresponding indeterminate, of any graph on $n$ vertices and $m$ edges.

An exact upper bound on the size of minimal clique covers