Isomorphic factorisations. I. Complete graphs

Authors:
Frank Harary, Robert W. Robinson and Nicholas C. Wormald

Journal:
Trans. Amer. Math. Soc. **242** (1978), 243-260

MSC:
Primary 05-02; Secondary 05C99

MathSciNet review:
0545305

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Abstract: An isomorphic factorisation of the complete graph is a partition of the lines of into *t* isomorphic spanning subgraphs *G*; we then write , and . If the set of graphs is not empty, then of course . Our principal purpose is to prove the converse. It was found by Laura Guidotti that the converse does hold whenever or . We give a new and shorter proof of her result which involves permuting the points and lines of . The construction developed in our proof happens to give all the graphs in and . The Divisibility Theorem asserts that there is a factorisation of into *t* isomorphic parts whenever *t* divides . The proof to be given is based on our proof of Guidotti's Theorem, with embellishments to handle the additional difficulties presented by the cases when *t* is not relatively prime to *p* or .

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DOI:
https://doi.org/10.1090/S0002-9947-1978-0545305-1

Keywords:
Factorisations,
complete graphs

Article copyright:
© Copyright 1978
American Mathematical Society