Adjacent vertex distinguishing total coloring of new families of graphs

Let [Formula: see text] be a positive integer. An adjacent vertex distinguishing total [Formula: see text]-coloring [Formula: see text] of a graph [Formula: see text] is a proper total [Formula: see text]-coloring of [Formula: see text] such that no pair of adjacent vertices has the same set of colors, where the set of colors at a vertex [Formula: see text] is [Formula: see text]. The minimum number of colors required for an adjacent vertex distinguishing total coloring of [Formula: see text] is called the adjacent vertex distinguishing total chromatic number and is denoted by [Formula: see text]. In this paper, we determined the exact value of [Formula: see text] for some families of graphs with [Formula: see text] like cube connected path, cube connected cycle, odd petal graph and [Formula: see text]-double star-based cubic graph and these results support Hulgan’s conjecture.