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Let G = (V , E) be a graph and f : (V ∪E) → [k] be a proper total k-coloring of G. We say that f is an adjacent vertex-distinguishing total coloring if for any two adjacent vertices, the set of colors appearing on the vertex and incident edges are different. We call the smallest k for which such a coloring of G exists the adjacent vertex-distinguishing total chromatic number, and denote it by χ at (G). Here we provide short proofs for an upper bound on the adjacent vertex-distinguishing totaldoi:10.1016/j.disc.2008.06.002 fatcat:i3yr6edw3rfjnjeavjwofi5xdi