关键词: 双向2重迹, 反射数, Betti亏数, 上可嵌入, 最大亏格

Abstract:

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.

Key words: Bidirectional double tracing, Retracing number, Betti , deficiency number, Upper embeddable, Maximum genus