For the purpose of analysing bipartite graphs (hereinafter called simply
graphs) the concept of an exterior covering is introduced. In terms of this
concept it is possible in a natural way to decompose any graph into two parts, an
inadmissible part and a core. It is also possible to decompose the core into
irreducible parts and thus obtain a canonical reduction of the graph. The concept
of irreducibility is very easily and naturally expressed in terms of exterior
coverings. The role of the inadmissible edges of a graph is to obstruct certain
natural coverings of the graph.
Euler (6) in 1782 first studied orthogonal latin squares. He showed the existence of a pair of orthogonal latin squares for all odd n and conjectured their non-existence for n = 2(2k + 1). MacNeish (8) in 1921 gave a construction of n — 1 mutually orthogonal latin squares for n = p with p prime and of n(v) mutually orthogonal squares of order v wherewith p1 p2, … , Pr being distinct primes and
In [1] G. Birkhoff stated an algorithm for expressing a doubly stochastic matrix as an average of permutation matrices. In this note we prove two graphical lemmas and use these to find an upper bound for the number of permutation matrices which the Birkhoff algorithm may use.A doubly stochastic matrix is a matrix of non-negative elements with row and column sums equal to unity and is there - fore a square matrix. A permutation matrix is an n × n doubly stochastic matrix which has n2-n zeros and consequently has n ones, one in each row and one in each column. It has been shown by Birkhoff [1],Hoffman and Wielandt [5] and von Neumann [7] that the set of all doubly stochastic matrices, considered as a set of points in a space of n2 dimensions constitute the convex hull of permutation matrices.
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