arXiv 2201.027812022-01-08math.CO

图的邻和可区别全染色的一个推广

An extension on neighbor sum distinguishing total coloring of graphs

Jing-zhi Chang, Chao Yang, Zhi-xiang Yin, Bing Yao

备注：: arXiv admin note: text overlap with arXiv:2107.00424

摘要：

设$f: V(G)\cup E(G)\rightarrow \{1,2,\dots,k\}$是$G$的非真全$k$ -染色.定义全着色的权重函数为$$ϕ(x)=f(x)+\sum\limits_{e\ni x}f(e)+\sum\limits_{y\in N(x)}f(y),$$，其中$N(x)=\{y\in V(G)|xy\in E(G)\}$。如果对任意边$xy\in E(G)$有$ϕ(x)\neq ϕ(y)$，则称$f$为$G$的邻满和可区别全$k$ -染色. $G$具有这种着色的最小值$k$被称为$G$的近邻全和可区别全色数，并由fgndi $_{\sum}(G)$表示。该染色是邻和可区别非正常全染色的推广。本文猜想对任意阶数至少为3的连通图$G$，fgndi $_{\sum}(G)\leq 3$是连通的.证明了该猜想对（i）路和圈，（ii）3-正则图，（iii）星，完全图，树，超立方体，二部图和完全$r$ -部图都成立.特别地，完全图可以达到上述猜想的上界。

英文摘要：

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.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

摘要：

英文摘要：

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