Abstract: Firstly,  according to the structural characteristics of Snark graphs, we constructed two classes of 3-regular graphs based on Double Star and Cross. Secondly, we studied the  problem of neighbor full sum distinguishing total coloring of four classes of 3-regular graphs by exhaustive coloring method and combinatorial analysis, and obtained that the neighbor full sum distinguishing total chromatic numbers for these graphs are all 2.

Key words: non-proper total coloring, neighbor full sum distinguishing total coloring, neighbor full sum distinguishing total chromatic number, 3-regular graph