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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

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