双向2 重迹与图的最大亏格

设G为连通图且L是G的一条双向2 重迹. 作者引入G的一个新参数, 称之为G的反射数,并用ε(G)表示. 反射数ε(G)由如下式子给出:ε(G)=min〖DD(X〗L〖DD)〗ε(G, L), 这里ε(G, L)是G的关于L的反射数,且“min”取遍G的所有双向2 重迹L然后, 对于3 正则图G, 作者证明了G的反射数ε(G)与G的最大亏格γ\-M(G)密切相关,具体地, ε(G)=2γ\-M(G)-β(G), 其中β(G)是G的圈秩数. 同时, 作者给出一个与ε(G)的值有关的G的特征结构. 这些可视为Thomassen C的有关结果的进一步补充.

Bidirectional Double Tracings and Maximum Genus

Let G be a connected graph and L be a bidirectional double tracing of G. The authors first introduce a new invarint of G, which is called the  retracing number and denoted by ε(G).  The definition  of ε(G) is  given as follows: ε(G)=min〖DD(X〗L〖DD)〗 ε(G, L), where  ε(G, L) is the number of retracings in L, and the minimum ranges over all bidirectional double tracings of G. Then, for a connected 3 regular graph G  the authors prove that ε(G) is closely related to the maximum genus γ\-M(G) of G, namely ε(G), equals to the value 2γ\-M(G)-β(G) where β(G) is the rank number of G.  Also the authors  provide an instructural characterization on thegraph G  according to the value ε(G). Thus these  may be viewed as  some  generalizations  of  Thomassen C's results.