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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

On the problem of partitioning $ \{1,\,2,\,\cdots ,\,n\}$ into subsets having equal sums


Authors: H. Joseph Straight and Paul Schillo
Journal: Proc. Amer. Math. Soc. 74 (1979), 229-231
MSC: Primary 05C38; Secondary 10A45
MathSciNet review: 524291
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Abstract: Let N denote the set of natural numbers and let $ {Z_n} = \{ 1,2, \ldots ,n\} $. For S a finite subset of N, let $ \sigma S$ denote the sum of the elements in S. Then $ \sigma {Z_n} = n(n + 1)/2$. Suppose $ n(n + 1) = 2st$, where s and t are integers and $ t \geqslant n$. We show that $ {Z_n}$ can be partitioned into $ {T_1} \cup {T_2} \cup \ldots \cup {T_s}$ such that $ \sigma {T_i} = t$, for $ 1 \leqslant i \leqslant s$. Such a partition is called an (s, t)-partition of $ {Z_n}$.

A graph G having $ n(n + 1)/2$ edges is said to be path-perfect if the edge set of G can be partitioned as $ {E_1} \cup {E_2} \cup \ldots \cup {E_n}$ so that $ {E_i}$ induces a path of length i, for $ 1 \leqslant i \leqslant n$. Suppose p and n are positive integers and r is an even positive integer with $ p \geqslant r + 1$ and $ pr = n(n + 1)$. The existence of an (r/2, p)-partition of $ {Z_n}$ is used to show the existence of an r-regular path-perfect graph G having p vertices and $ n(n + 1)/2$ edges.


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DOI: http://dx.doi.org/10.1090/S0002-9939-1979-0524291-0
PII: S 0002-9939(1979)0524291-0
Article copyright: © Copyright 1979 American Mathematical Society