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
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