Abstract:

Let $f: V(G)\cup E(G)\rightarrow \{1, 2, \cdots, k\}$ be a proper $k$-total coloring of $G$. Set $\phi(x)=f(x)+\sum\limits_{e\ni x}f(e)+\sum\limits_{y\in N(x)}f(y)$, where $N(x)=\{y\in V(G)|xy\in E(G)\}$. If $\phi(u)\neq \phi(v)$ for any edge $uv\in E(G)$, then $f$ is called a $k$-neighbor full sum distinguishing total coloring of $G$. The smallest value $k$ for which $G$ has such a coloring is called the neighbor full sum distinguishing total chromatic number of $G$ and denoted by $ftndi_{\sum}(G)$. In this paper, we obtain this parameter for paths, cycles, stars, wheels, complete bipartite graphs, complete graphs and trees. Meanwhile, we conjecture that the neighbor full sum distinguishing total chromatic number of $G(\neq K_2)$ is not more than $\Delta(G)+2$.

Key words: proper total coloring, distinguishing coloring, neighbor full sum distinguishing total coloring, neighbor full sum distinguishing total chromatic number