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 3regular 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.
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