Abstract
Suppose the simple graph G(V, E) is at least 2nd-order connected. We study the adjacent vertex-distinguishing equitable-total coloring of the join graph Pm ∨ Fm which belongs to the graph G(V, E). By constructing the total coloring of Pm ∨ Fm , we obtain the adjacent vertex-distinguishing equitable-total chromatic number of it, which proves to satisfy the adjacent vertex-distinguishing equitable-total coloring conjecture.
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