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An Enumeration Problem Related to the Number of Labelled Bi-Coloured Graphs

  • C. Y. Lee (a1)

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We will consider the following enumeration problem. Let A and B be finite sets with α and β elements in each set respectively. Let n be some positive integer such that nαβ. A subset S of the product set A × B of exactly n distinct ordered pairs (ai, bj) is said to be admissible if given any aA and bB, there exist elements (ai, bj) and (ak, bl) (they may be the same) in S such that ai = a and bl = b. We shall find here a generating function for the number N(α, β n) of distinct admissible subsets of A × B and from this generating function, an explicit expression for N(α, β n). In obtaining this result, the idea of a cut probability is used. This approach in a problem of enumeration may be of interest.

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References

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1. Polya, G., Kombinatorische Anzahlbestimmungen fur Gruppen, Graphen, una chemishe Verbindungen, Acta Math., 68 (1937), 145-254.
2. Harary, F., On the number of bi-colored graphs, Pac. J. Math., 8 (1958), 743-755.
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An Enumeration Problem Related to the Number of Labelled Bi-Coloured Graphs

  • C. Y. Lee (a1)

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