On the adjacent vertex-distinguishing total colorings of some cubic graphs

Suppose \(G = (V, E)\) is a simple graph and \(f: (V\cup E) → {1,2,…, k}\) is a proper total k-coloring of G. Let \(C(u) = {f(u)} \cup {f(uv): uv \in E(G)}\) for each vertex u of G. The coloring f is said to be an adjacent vertex-distinguishing total coloring of G if \(C(u) \neq C(v)\) for every \(uv \in E(G)\). The minimum k for which such a chromatic number of G, and is denoted by \(X_at (G)\). This paper considers three types of cubic graphs: a specific family of cubic hasmiltonian graphs, snares and Generalized Petersen graphs. We prove that these cubic graphs have the same adjacent vertex-distinguishing total chromatic number 5. This is a step towards a problem that whether the bound \(X_at (G) ≤ 6\) is sharp for a graph G with maximum degree three.