The Adjacent Vertex Distinguishing Total Chromatic Number of Graphs

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). In this paper, we show that χ_at(K₁₉ - E(C₄)) = 20 and χ_at (K₂₁-E(C₄)) = 22.