Simple minimum coverings ofK _n with copies ofK ₄ −e

Summary

AK ₄−e design of ordern is a pair (S, B), whereB is an edge-disjoint decomposition ofK _n (the complete undirected graph onn vertices) with vertex setS, into copies ofK ₄−e, the graph on four vertices with five edges. It is well-known [1] thatK ₄−e designs of ordern exist for alln ≡ 0 or 1 (mod 5),n ≥ 6, and that if (S, B) is aK ₄−e design of ordern then |B| =n(n − 1)/10.

Asimple covering ofK _n with copies ofK ₄−e is a pair (S, C) whereS is the vertex set ofK _n andC is a collection of edge-disjoint copies ofK ₄−e which partitionE(K_n)⋃P, for some$P \subseteq E(K_n )$. Asimple minimum covering ofK _n (SMCK _n) with copies ofK ₄−e is a simple covering whereP consists of as few edges as possible. The collection of edgesP is called thepadding. Thus aK ₄−e design of ordern isSMCK _n with empty padding.

We show that forn ≡ 3 or 8 (mod 10),n ≥ 8, the padding ofSMCK _n consists of two edges and that forn ≡ 2, 4, 7 or 9 (mod 10),n ≥ 9, the padding consists of four edges. In each case, the padding may be any of the simple graphs with two or four edges respectively. The smaller cases need separate treatment:SMCK ₅ has four possible paddings of five edges each,SMCK ₄ has two possible paddings of four edges each andSMCK ₇ has eight possible paddings of four edges each.

The recursive arguments depend on two essential ingredients. One is aK ₄−e design of ordern with ahole of sizek. This is a triple (S, H, B) whereB is an edge-disjoint collection of copies ofK ₄−e which partition the edge set ofK _n\K_k, whereS is the vertex set ofK _n, and$H \subseteq S$ is the vertex set ofK _k. The other essential is acommutative quasigroup with holes. Here we letX be a set of size 2n ≥ 6, and letπX = {x ₁, x₂, ..., x_n} be a partition ofX into 2-element subsets, calledholes of size two. Then a commutative quasigroup with holesπX is a commutative quasigroup (X, ∘) such that for each holex _i ∈ πX, (x_i, ∘) is a subquasigroup. Such quasigroups exist for every even order 2n ≥ 6 [4].