On \(k\)-complementing permutations of cyclically \(k\)-complementary graphs (Q1916377)

Let \(K_n\) be a complete graph and \(H_1,H_2,\dots,H_k\) its isomorphic factorization where \(k>1\) and \(k\) divides \({1\over 2n} (n-1)\). Let \(\beta\) be a permutation on \(V(K_n)\) such that (by a suitable rearrangement of \(H_i\)) \(\beta: V(H_i)\to V(H_{i+1})\) is an isomorphism for \(i=1,2,\dots,k-1\). Then any isomorphical image of a factor \(H_i\) is said to be a cyclically \(k\)-complementary graph \(G(k,n)\). The permutation \(\beta\) is called a \(k\)-complementing permutation of \(G(k,n)\). The author gives necessary and sufficient conditions to be a power \(\beta^t\) of a \(k\)-complementing permutation \(\beta\) of \(G(k,n)\): (1) an automorphism of \(G(k,n)\), and (2) a \(k\)-complementing permutation of \(G(k,n)\). Further, he gives some special results which follow from a decomposition \(\beta= \beta_1\beta_2\cdots\beta_r\) of a \(k\)-complementing permutation \(\beta\) of \(G(k,n)\).