设$f:V(G)\cup E(G)\rightarrow \{1, 2, \cdots, k\}$是图$G$的一个正常$k$-全染色。令$\phi(x)=f(x)+\sum\limits_{e\ni x}f(e)+\sum\limits_{y\in N(x)}f(y)$, 其中$N(x)=\{y\in V(G)|xy\in E(G)\}$。对任意的边$uv\in E(G)$, 若有$\phi(u)\neq \phi(v)$成立, 则称$f$是图$G$的一个邻点全和可区别$k$-全染色。图$G$的邻点全和可区别全染色中最小的颜色数$k$叫做$G$的邻点全和可区别全色数, 记为$ftndi_{\sum}(G)$。本文确定了路、圈、星、轮、完全二部图、完全图以及树的邻点全和可区别全色数, 同时猜想: 简单图$G(\neq K_2)$的邻点全和可区别全色数不超过$\Delta(G)+2$。

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 $\Delta(G)+2$.