The Dimension Of The Space Spanned By Cycles With Constraint Conditions

This article has two parts. The first part, showed in Chapter Two, mainly studies the dimension of the space spanned by Hamilton cycles in complete graphs. We start with the simplest situation, then adding some constraint conditions to the cycles, which include:(1) Hamilton cycles in complete graphs;(2) Hamilton cycles avoiding given matching in complete graphs;(3) Hamilton cycles including given matching in complete graphs. The study is carried on by the method of construction.The second part, showed in Chapter Three, focuses on the dimension of the space spanned by directed cycles in strongly connected oriented graph. The space here is spanned by the directed cycles without the direction, i.e. the normal cycles. Then we turn to the property of the minimum directed cycle base. One article[7] proved that the structure of the minimum cycle base(MCB) is unique, which means there is a mapping between any two minimum cycle bases. A cycle in one MCB is mapped to a cycle with the same length in the other MCB. We try to show that the structure of the minimum directed cycle base in strongly connected oriented graphs is unique as well, along with the statement and proof of the uniqueness in undirected graph.