Complete bipartite decompositions of crowns, with applications to complete directed graphs

Abstract

For an integer n≥3, the crown S ⁰_n is defined to be the graph with vertex set {a ₁, a ₂, ..., a _n , b ₁, b ₂, ..., b _n } and edge set {a _i b _j : 1≤i,j≤n,i≠j}. We consider the decomposition of the edges of S ⁰_n into the complete bipartite graphs, and obtain the following results.

The minimum number of complete bipartite subgraphs needed to decompose the edges of S ⁰_n is n.

The crown S ⁰_n has a K _l,m -decomposition (i.e., the edges of S ⁰_n can be decomposed into subgraphs isomorphic to K _l,m ) if n=λlm+1 for some positive integers λ, l, m. Furthermore, the l-part and m-part of each member in this decomposition can be required to be contained in {a ₁, a ₂, ..., a _n } and {b ₁,b ₂, ..., b _n }, respectively.

Every minimum complete bipartite decomposition of S ⁰_n is trivial if and only if n=p+1 where p is prime (a complete bipartite decomposition of S ⁰_n that uses the minimum number of complete bipartite subgraphs is called a minimum complete bipartite decomposition of S ⁰_n and a complete bipartite decomposition of S ⁰_n is said to be trivial if it consists of either n maximal stars with respective centers a ₁, a ₂, ..., a _n , or n maximal stars with respective centers b ₁, b ₂, ..., b _n ).

The above results have applications to the directed complete bipartite decomposition of the complete directed graphs.