论文标题
邻居总和区分图的总彩色的扩展
An extension on neighbor sum distinguishing total coloring of graphs
论文作者
论文摘要
令$ f：v（g）\ cup e（g）\ rightArrow \ {1,2，\ dots，k \} $是$ g $的非proper $ k $ - 颜色。将重量功能定义为$$ ϕ（x）= f（x）+\ sum \ limits_ {e \ ni x} f（e）+\ sum \ limits_ {y \ in n（x）} f（y），$ n（y），$ n（x）= \ n（x）= \ {y in v（y in v in v（g） @ xy e（x y（g）如果对于任何边缘$ xy \ e（g）$中的任何边缘$ xy \ $ ϕ（x）\ neq ϕ（y）$，则$ f $称为邻居全额分别区分$ g $的总$ k $ - 颜色。 $ g $具有这样的着色的最小值$ k $称为邻居总和区分$ g $的总色数，并用fgndi $ _ {\ sum}（g）$表示。着色是邻居总和区分非操作器总彩色的扩展。在本文中，我们猜测FGNDI $ _ {\ sum}（g）\ leq 3 $对于任何连接的订单的订单$ g $至少三个。我们证明了（i）路径和周期的猜想是正确的。 （ii）3个规则图和（iii）星星，完整的图形，树，超振管，两部分图和完整的$ r $ - 分段图。特别是，完整的图可以实现上述猜想的上限。
Let $f: V(G)\cup E(G)\rightarrow \{1,2,\dots,k\}$ be a non-proper total $k$-coloring of $G$. Define a weight function on total coloring as $$ϕ(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 $ϕ(x)\neq ϕ(y)$ for any edge $xy\in E(G)$, then $f$ is called a neighbor full sum distinguishing total $k$-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 fgndi$_{\sum}(G)$. The coloring is an extension of neighbor sum distinguishing non-proper total coloring. In this paper we conjecture that fgndi$_{\sum}(G)\leq 3$ for any connected graph $G$ of order at least three. We prove that the conjecture is true for (i) paths and cycles; (ii) 3-regular graphs and (iii) stars, complete graphs, trees, hypercubes, bipartite graphs and complete $r$-partite graphs. In particular, complete graphs can achieve the upper bound for the above conjecture.
