三次图和4-正则图的总顶点不规则强度

中文摘要 AI 辅助

设$G$为一个图，$k$为一个正整数。$G$的一个总$k$-标号将每个顶点和每条边分配一个来自集合$\{1,\ldots,k\}$的标签。一个顶点的权重是其标签与其关联边的标签之和。如果所有顶点的权重互不相同，则称该总标号是顶点不规则的。总顶点不规则强度$\text{tvs}(G)$是使得$G$具有一个顶点不规则总$k$-标号的最小$k$。对于有$n$个顶点的$r$-正则图$G$，一个计数论证给出$\text{tvs}(G)\ge\lceil(n+r)/(r+1)\rceil$。Nurdin、Baskoro、Salman和Gaos的一个猜想在正则图上的限制断言该下界可以达到。我们证明了该断言对于三次图和$4$-正则图成立。我们还表明，对于每个固定的$r\ge2$，一个关于指定度频率的最近定理蕴含了该断言对所有足够大的$r$-正则图成立。

英文摘要

Let $G$ be a graph and $k$ be a positive integer. A total $k$-labeling of $G$ assigns to each vertex and each edge a label from $\{1,\ldots,k\}$. The weight of a vertex is the sum of its label and the labels of its incident edges. A total labeling is vertex irregular if all vertex weights are distinct. The total vertex irregularity strength $\text{tvs}(G)$ is the smallest $k$ for which $G$ has a vertex irregular total $k$-labeling. For an $r$-regular graph $G$ on $n$ vertices, a counting argument gives $\text{tvs}(G)\ge\lceil(n+r)/(r+1)\rceil$. The restriction of a conjecture of Nurdin, Baskoro, Salman, and Gaos to regular graphs asserts that this bound is attained. We prove this assertion for cubic and $4$-regular graphs. We also show that, for every fixed $r\ge2$, a recent theorem on prescribed degree frequencies implies the assertion for all sufficiently large $r$-regular graphs.

发表机构

• Auburn University（奥本大学）

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