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

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.