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Application of analytic method and Constructing the Smarandachely adjacent vertex distinguishing E-total coloring function, the Smarandachely adjacent vertex distinguishing E-total coloring of two kinds of 3-regular graphs are mainly discussed, and the Smarandachely adjacent vertex distinguishing E-total chromatic number of join graph are obtained. The Smarandachely adjacent vertex distinguishing E-total coloring conjecture is further validated.

The Smarandachely adjacent-vertex total chromatic number of graph $G$ is the smallest positive integer $k$ for which $G$ has a proper total $k$-coloring such that for any pair of adjacent vertices $u,v$, the set of colors appearing at vertex $u$ and its incident edges is not a subset of the set of colors appearing at vertex $v$ and its incident edges. In this paper, a kind of Generalized 3-regular Ring Graph is defined, and the Smarandachely adjacent-vertex total chromatic numbers is discussed by using construct coloring function. Furthermore, the conjecture of Smarandachely adjacent-vertex total coloring of 3-regular graph is checked.