Minimum-excess coverings of complete graphs by 3-, 4-, and 5-cliques

Abstract

We study two-level optimal coverings of the complete graph Kv by cliques of orders 3, 4, and 5. The first level minimizes the excess, namely the number of repeated edge occurrences, and the second level minimizes the number of cliques among coverings with minimum excess. We first isolate the two-size problem for triangles and 4-cliques, where the main congruence and local-degree methods already appear and where nonzero excess is unavoidable for one residue class of v. This motivates the passage to quintuples. For {K3 , K4 , K5 } coverings we determine the minimum excess for every v, use an edge-count reduction for the secondary optimization, obtain exact values in ten residue classes modulo 20, and give bounds for the remaining classes.