# Non-uniform covering array with symmetric forbidden edge constraints

## TL;DR

This paper investigates the properties of covering arrays with forbidden edge constraints, showing that even highly symmetric constraints can lead to non-uniform optimal arrays, challenging previous assumptions about their uniformity.

## Contribution

It proves the existence of a symmetric constraint graph where the optimal covering array is non-uniform, providing a counterexample to the presumed universality of uniformity.

## Key findings

- Highly symmetric constraints can force non-uniform optimal arrays
- Counterexample to the conjecture that optimal arrays are always uniform
- Optimal covering arrays may not be uniform under certain symmetric constraints

## Abstract

It has been conjectured that whenever an optimal covering array exists there is also a uniform covering array with the same parameters and this is true for all known optimal covering arrays. When used as a test suite, the application context may have pairs of parameters that must be avoided and Covering arrays avoiding forbidden edges (CAFE) are a generalization accommodating this requirement. We prove that there is an arc-transitive, highly symmetric constraint graph where the unique optimal covering array avoiding forbidden edges is not uniform. This does not refute the conjecture but it does show that placing even highly symmetric constraints on covering arrays can force non-uniformity of optimal arrays.