【摘要】 1974年Dewdney提出了n维复形上的(m,n)树的概念和关于(m,n)树的两个猜想。本文解决了这两个猜想。指出它们是不成立的,同时证明了纯粹复形是(m,n)树的一个充要条件(定理1)。它的充分条件是不能再减弱的。解决上述问题的方法是引进复形K上的(m,n)关联二分图Ka_ma_n和利用两个定理及其推论。一个定理讨论了K的(m,n)连通与Ka_ma_n的连通的关系;另一个讨论了K中无(m,n)圈与Ka_ma_n无圈的关系。更多还原

【Abstract】 Dewdney introduced the concept of a (m,n)-tree on a n-complex and proposed two conjectures about the necessary and sufficient conditions for a pure n-complex to be a (m,n)-tree in 1974. In present paper his conjectures are distrored by counterexampls. In addition, a theorem which states acorrect necessary and sufficient condition for a pure n-complex to be a (m,n)-tree is proved and it is also shown that the sufficient condition can not be weakened further.The method used in present paper is to define a (m, n ) incident bipartite graph Ka_ma_n for a given complex K and to reduce the (m,n) connectivity of K to the connectivity of Ka_ma_n and to reduce the non-existence of (m,n) circuits in K to the non-existence of ciruits in Ka_ma_n.更多还原