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

## Abstract

For an integer *n*≥3, the crown *S* _{n} ^{0} 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* _{n} ^{0} into the complete bipartite graphs, and obtain the following results.

The minimum number of complete bipartite subgraphs needed to decompose the edges of *S* _{n} ^{0} is *n*.

The crown *S* _{n} ^{0} has a *K*_{ l,m }-decomposition (i.e., the edges of *S* _{n} ^{0} 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* _{n} ^{0} is trivial if and only if *n=p*+1 where *p* is prime (a complete bipartite decomposition of *S* _{n} ^{0} that uses the minimum number of complete bipartite subgraphs is called a minimum complete bipartite decomposition of *S* _{n} ^{0} and a complete bipartite decomposition of *S* _{n} ^{0} 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.

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