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Some Addition Theorems of Group Theory with Applications to Graph Theory

Published online by Cambridge University Press:  20 November 2018

H. P. Yap
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Let G be an additive group with nonempty subsets S and T. Let S ± T = {s ± t; s ∊ S, t ∊ T}, let S be the set complement of S in G, and let |S| be the cardinality of S. We abbreviate {f} where f ∊ G to f. If S + S and S have no element in common, then we say that S is a sum-free set in G or that S is sum-free in G. If S is a sum-free set in G and if for every sum-free set T in G, |S| ≧ |T|, then S is said to be a maximal sum-free set in G. We denote by λ(G) the cardinality of a maximal sum-free set in G. We say that S is in arithmetic progression having difference d if S = {s, s + d, …, s + nd} for some s, d ∊ G and some integer n ≧ 0.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 22 , Issue 6 , 01 December 1970 , pp. 1185 - 1195
Copyright
Copyright © Canadian Mathematical Society 1970

References

1. Berge, C., Theory of graphs and its applications (Methuen, London, 1962).Google Scholar
2. Diananda, P. H. and Yap, H. P., Maximal sum-free sets of elements of finite groups, Proc. Japan Acad. 45 (1969), 1–5.Google Scholar
3. Mann, H. B., Addition theorems (Interscience, New York, 1965).Google Scholar
4. Mann, H. B. and Olson, J. E., Sums of sets in the elementary abelian group of type (p, p), J. Combinatorial Theory 2 (1967), 275–284.Google Scholar
5. Sabidussi, G., On a class of fix-point-free graphs, Proc. Amer. Math. Soc. 9 (1958), 800–804.Google Scholar
6. Teh, H. H. and Yap, H. P., Some construction problems of homogeneous graphs, Bull. Math. Soc. Xanyang Univ. 1964, 164–196.Google Scholar
7. Turner, J., Point-symmetric graphs with a prime number of points, J. Combinatorial Theory 3 (1967), 136–145.Google Scholar
8. Yap, H. P., The number of maximal sum-free sets in Cp, Nanta Math. 2 (1968), 68–71.Google Scholar
9. Yap, H. P., A class of point-symmetric graphs, Nanta Math. 8 (1969), 100–109.Google Scholar