Let Formula: see text be a positive integer. An adjacent vertex distinguishing total Formula: see text-colouring Formula: see text of a graph Formula: see text is a proper total Formula: see text-colouring of Formula: see text such that no pair of adjacent vertices have the same set of colours, where the set of colours at a vertex Formula: see text is Formula: see text. The minimum number of colours required for an adjacent vertex distinguishing total colouring of Formula: see text is called the adjacent vertex distinguishing total chromatic number and is denoted by Formula: see text. In this article, 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.
Selvi et al. (Thu,) studied this question.