论文标题
全图的笛卡尔产品总计$ k $ - 域名
Total $k$-domination in Cartesian product of complete graphs
论文作者
论文摘要
令$ g =（v，e）$为有限的无向图。如果$ v $ $ v $中的$ v $中的$ s $ s $ thetices在$ v $中的每个顶点均与至少$ k $ s $中的至少$ k $ dertices相邻，则为$ k $。总$ k $ - 域名，$γ_{kt}（g）$，是$ g $中$ k $ domination设置的最小基数。在这项工作中，我们研究了两个完整图的笛卡尔产品的总$ k $域名数量，这是两个图的笛卡尔产品的总$ k $ domination的下限。我们获得了两个完整图的笛卡尔产品的$ K $ domination数量的新的下层和上限。由于我们发现的边界，获得了一些渐近行为。特别是，我们获得了$ \ displaystyle \ liminf_ {n \ to \ infty} \ frac {γ_{kt}（g \ box h）} {n} {n} \ leq 2 \，\左（\ left \ lceil \ frac {k} {2} \ right \ rceil^{ - 1}+\ left \ left \ lfloor \ frac {k+4} {2} {2} {2} {2} \ right \ right \ rfloor^rfloor^{ - 1} { - 1} \ right）我们还证明，当且仅当$ k $都达到时，才能达到平等。当$ g，h $均与完整的图相同时，$ k_n $，带有$ n $ vertices时，平等成立。此外，我们获得了$ 2 $ 2 $ - 域名的载货品的封闭公式，这些笛卡尔产品的两个完整图表的任何订单都可以。此外，我们证明，对于$ k = 3 $，上面的不平等是可以改进到$ \ displayStyle \ liminf_ {n \ to \ infty}γ_{3T}（k_n \ box k_n）/n \ leq 11/5 $。
Let $G=(V,E)$ be a finite undirected graph. A set $S$ of vertices in $V$ is said to be total $k$-dominating if every vertex in $V$ is adjacent to at least $k$ vertices in $S$. The total $k$-domination number, $γ_{kt}(G)$, is the minimum cardinality of a total $k$-dominating set in $G$. In this work we study the total $k$-domination number of Cartesian product of two complete graphs which is a lower bound of the total $k$-domination number of Cartesian product of two graphs. We obtain new lower and upper bounds for the total $k$-domination number of Cartesian product of two complete graphs. Some asymptotic behaviors are obtained as a consequence of the bounds we found. In particular, we obtain that $\displaystyle\liminf_{n\to\infty}\frac{γ_{kt}(G\Box H)}{n}\leq 2\,\left(\left\lceil\frac{k}{2}\right\rceil^{-1}+\left\lfloor\frac{k+4}{2}\right\rfloor^{-1}\right)^{-1}$ for graphs $G,H$ with order at least $n$. We also prove that the equality is attained if and only if $k$ is even. The equality holds when $G,H$ are both isomorphic to the complete graph, $K_n$, with $n$ vertices. Furthermore, we obtain closed formulas for the total $2$-domination number of Cartesian product of two complete graphs of whatever order. Besides, we prove that, for $k=3$, the inequality above is improvable to $\displaystyle\liminf_{n\to\infty} γ_{3t}(K_n\Box K_n)/n \leq 11/5$.
