A note on the adjacent vertex distinguishing total chromatic number of graphs

AbstractAn adjacent vertex distinguishing total coloring of a graph G is a proper total coloring of G such that any pair of adjacent vertices have different sets of colors. The minimum number of colors needed for such a total coloring of G is denoted by χa″(G). In this note, we show that χa″(G)≤2Δ for any graph G with maximum degree Δ≥3

On adjacent vertex distinguishing total coloring of quadrilateral snake

In this paper, we prove the existence of the adjacent vertex distinguishing total coloringnof quadrilateral snake, double quadrilateral snake, alternate quadrilateral snake and double alternate quadrilateral snake in detail. Also, we present an algorithm to obtain the adjacent vertex distinguishing total coloring of these quadrilateral graph family. The minimum number of colors required to give an adjacent vertex distinguishing total coloring (abbreviated as AVDTC) to the graph G is denoted by avt(G)