对偶图K_(n,n)+I的循环(m_1,m_2,…,m_r)-圈分解

Cyclically Even(m_1,m_2,...,m_r)-Cycle Decompositions of K_(n,n)+I

mi(1≤i≤r)为偶数且r∑(i=1)mi=2k(k≥1).Kn,n为偶图,I为Kn,n的一因子.证明了Kn,n+I可分解为(m1,m2,…,mr)-圈的充分必要条件为2k│n(n+1)且n为奇数.进一步,Kn,n+I可分解为循环的(m1,m2,…,mr)-圈充分必要条件为2k=n+1且n为奇数.

Let mi（1≤i≤r） be positive integer, ^r∑i=1mi=2^k（k≥1）, Kn,n be a complete bipartite graph and I be a one-factor of Kn,n. It is proved that Kn,n＋I can be decomposed into （m1,m2,…,mr）-Cycles if and only if 2^k｜n（n＋1） and n is odd. Moreover, Kn,n＋I can be cyclically decomposed into （m1,m2,…,mr）-Cycles if and only if n＋1=2^k and n is odd.