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

  • Chiang Lin
  • Jenq-Jong Lin
  • Tay-Woei Shyu
Graph Theory
Part of the Lecture Notes in Computer Science book series (LNCS, volume 1120)

Abstract

For an integer n≥3, the crown S n 0 is defined to be the graph with vertex set {a1, a2, ..., a n , b1, b2, ..., b n } and edge set {a i b j : 1≤i,jn,ij}. 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 {a1, a2, ..., a n } and {b1,b2, ..., 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 a1, a2, ..., a n , or n maximal stars with respective centers b1, b2, ..., b n ).

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

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Copyright information

© Springer-Verlag Berlin Heidelberg 1996

Authors and Affiliations

  • Chiang Lin
    • 1
  • Jenq-Jong Lin
    • 1
  • Tay-Woei Shyu
    • 1
  1. 1.Department of MathematicsNational Central UniversityChung-LiTaiwan, R.O.C.

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