# An exact upper bound on the size of minimal clique covers

## TL;DR

This paper determines the exact maximum size of the minimal clique cover for any graph with given vertices and edges, which relates to the minimal alphabet size in indeterminate string representations.

## Contribution

It provides an exact upper bound on the size of minimal clique covers for graphs with specified vertices and edges, addressing a problem from prior research.

## Key findings

- Computed the exact maximum minimal clique cover size for given graph parameters
- Established a precise upper bound related to graph structure and clique covers
- Linked the bound to minimal alphabet size in indeterminate string representations

## Abstract

Indeterminate strings have received considerable attention in the recent past; see for example Christodoulakis et al 2015 and Helling et al 2017. 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 Helling et al 2017: compute $\Theta_n(m)$, 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.