Total and adjacent vertex-distinguishing total chromatic numbers of augmented cubes

A total coloring of a graph G is a coloring of both the edges and the vertices. A total coloring is proper if no two adjacent or incident elements receive the same color. An adjacent vertex-distinguishing total coloring h of a simple graph G = (V, E) is a proper total coloring of G such that H(u) not equal H(v) for any two adjacent vertices u and v, where H(u) = {h(wu)vertical bar wu is an element of E(G)} boolean OR {h(u)} and H(v) = {h(xv)vertical bar xv is an element of E(G)} boolean OR {h(v)}. The minimum number of colors required for a proper total coloring (resp. an adjacent vertex-distinguishing total coloring) of G is called the total chromatic number (resp. adjacent vertex-distinguishing total chromatic number) of G and denoted by chi(t)(G) (resp. chi(at)(G)). The Total Coloring Conjecture (TCC) states that for every simple graph G, Delta(G) + 1 <= chi(t)(G) <= Delta(G) + 2. G is called Type 1 (resp. Type 2) if chi(t)(G) = Delta(G) + 1 (resp. chi(t)(G) = Delta(G) + 2). In this paper, we prove that the augmented cubes AQ(n) is of Type 1 for n >= 4. We also consider the adjacent vertex-distinguishing total chromatic number of AQ(n), prove that chi(at)(AQ(n)) = Delta(AQ(n)) + 2 for n >= 3.