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Let  N  denote the set of natural numbers and let      Z n = { 1 , 2 , …  , n } {Z_n} = \{ 1,2, \ldots ,n\}      . For  S  a finite subset of  N , let      σ  S \sigma S       denote the sum of the elements in  S . Then      σ  Z n = n ( n + 1 ) / 2 \sigma {Z_n} = n(n + 1)/2      . Suppose      n ( n + 1 ) = 2 s t n(n + 1) = 2st      , where  s  and  t  are integers and      t ⩾  n t \geqslant n      . We show that      Z n {Z_n}       can be partitioned into      T 1 ∪  T 2 ∪  …  ∪  T s {T_1} \cup {T_2} \cup \ldots \cup {T_s}       such that      σ  T i = t \sigma {T_i} = t      , for      1 ⩽  i ⩽  s 1 \leqslant i \leqslant s      . Such a partition is called an ( s, t )-partition of      Z n {Z_n}      . A graph  G  having      n ( n + 1 ) / 2 n(n + 1)/2       edges is said to be path-perfect if the edge set of  G  can be partitioned as      E 1 ∪  E 2 ∪  …  ∪  E n {E_1} \cup {E_2} \cup \ldots \cup {E_n}       so that      E i {E_i}       induces a path of length  i , for      1 ⩽  i ⩽  n 1 \leqslant i \leqslant n      . Suppose  p  and  n  are positive integers and  r  is an even positive integer with      p ⩾  r + 1 p \geqslant r + 1       and      p r = n ( n + 1 ) pr = n(n + 1)      . The existence of an ( r /2,  p )-partition of      Z n {Z_n}       is used to show the existence of an  r -regular path-perfect graph  G  having  p  vertices and      n ( n + 1 ) / 2 n(n + 1)/2       edges.

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