General Vertex-Distinguishing Total Colorings of Complete Bipartite Graphs

Let G be a simple graph. A general total coloring f of G refers to a coloring of the vertices and edges of G. Let $C\left(x\right)$ be the set of colors of vertex x and edges incident with x under f. For a general total coloring f of G in which k colors are available, if $C\left(u\right)\ne C\left(v\right)$ for any two different vertices u and v in $V\left(G\right)$, then f is called a k-general vertex-distinguishing total coloring of G, or a k-GVDTC of G for short. The minimum number of colors required for a GVDTC of G is denoted by ${\chi }_{gvt}\left(G\right)$ and is called the general vertex-distinguishing total chromatic number, or the GVDT chromatic number of G for short. GVDTCs of complete bipartite graphs are studied in this paper.