Minimum covering by triples, quadruples and quintuples with minimum excess

TL;DR

This paper investigates optimal graph coverings using small cliques of sizes three to five, focusing on minimizing excess and determining minimal coverings for triples and quadruples, with generalizations provided.

Contribution

It introduces the concept of minimum covering with minimum excess and determines minimal coverings by triples and quadruples, extending to larger clique combinations.

Findings

Minimum covering by triples and quadruples with minimum excess is determined.

Generalizations to coverings with triples, quadruples, and quintuples are presented.

Theoretical bounds and optimal configurations are identified.

Abstract

This article explores a new type of optimal covering of a complete graph by small cliques of different sizes, namely the minimum covering with minimum excess. In particular, the minimum size of a covering by triples and quadruples with minimum excess is determined. Moreover, some generalisation onto minimum coverings by triples, quadruples and quintuples with minimum excess is presented.