Generalized covering designs and clique coverings

TL;DR

This paper introduces generalized covering designs that unify covering designs and arrays, providing bounds and construction methods, with interpretations via clique coverings of graphs.

Contribution

It defines a new class of combinatorial designs, establishes bounds, and offers optimal construction methods, extending the theory of covering designs.

Findings

Derived bounds on minimum sizes of the designs

Proposed construction methods with some proven optimality

Interpreted designs as clique coverings of graphs

Abstract

Inspired by the "generalized t-designs" defined by Cameron [P. J. Cameron, A generalisation of t-designs, Discrete Math. 309 (2009), 4835--4842], we define a new class of combinatorial designs which simultaneously provide a generalization of both covering designs and covering arrays. We then obtain a number of bounds on the minimum sizes of these designs, and describe some methods of constructing them, which in some cases we prove are optimal. Many of our results are obtained from an interpretation of these designs in terms of clique coverings of graphs.