Tutte关于3-连通图的结构定理表明: 每一个3-连通图都可由某个轮图(也是 Halin图)经顶点分裂逐步得到. 这表明了Halin图在图结构研究中的地位和作用. 首先研究得到了近正则Halin图的消圈数的上、下界并证明了上述界是紧的, 接着得到了最大度为k 或最小度为k 的 Halin图的消圈数所满足的界; 此外还研究了Halin图的点染色 问题, 给出了它的点色数定理的一个新证明.

According to the structural theorem of 3-connected graphs by Tutte, every 3-connected graph can be obtained by splitting vertices of some wheel which is Halin graph, which indicates that the study of the structure of Halin graph is important in graph structures. In this paper, firstly we dealt with the decycling number of the nearly k-regular Halin graphs, and we got the bidirectional inequality that the decycling numbers of nearly regular Halin graphs must satisfy, then we proved that the boundaries above are tight and got the boundaries of Halin graphs with the most biggest degree or the least degree k. At last, we gave a new proof to the theorem about the (vertex) coloring of Halin graphs.