图的邻点全和可区别全染色

设f:V(G)∪E(G)→{1,2,…,k}是图G的一个正常k-全染色。令■其中N(x)={y∈V(G)|xy∈E(G)}。对任意的边uv∈E(C),若有Φ(u)≠Φ(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f tndi∑(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(≠K2)的邻点全和可区别全色数不超过△(G)+2。

Neighbor full sum distinguishing total coloring of graphs

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 $\Delta(G)+2$.