arXiv 0712.26882007-12-17math.CO

立方性、盒性与顶点覆盖

Cubicity, Boxicity and Vertex Cover

L. Sunil Chandran, Anita Das, Chintan Shah

备注：: 12 pages

摘要：

一个$k$ -维盒是卡氏积$R_1 \times R_2 \times ... \times R_k$，其中每个$R_i$是实直线上的一个闭区间.图$G$的{ \it boxicity}表示为$box(G)$，是最小整数$k$，使得$G$是$k$维盒的集合的交集图。$k$维空间中的单位立方体或$k$ -立方体被定义为笛卡尔积$R_1 \times R_2 \times ... \times R_k$，其中每个$R_i$是$[a_i, a_{i}+1]$形式的实直线上的闭区间。$G$的{ \it cubicity}表示为$cub(G)$，是最小$k$，使得$G$是$k$ -立方体集合的交集图。本文证明了$G$的最小顶点覆盖的基数为$cub(G) \leq t + \left \lceil \log (n - t)\right\rceil - 1$，$box(G) \leq \left \lfloor\frac{t}{2}\right\rfloor + 1$为$box(G) \leq \left \lfloor\frac{t}{2}\right\rfloor + 1$，$n$为$G$的顶点数.我们还显示了这些上界的紧密性。F. S.罗伯茨在他关于盒性和立方性的开创性论文中证明了，对于一个图$G$，$box(G) \leq \left \lfloor\frac{n}{2} \right \rfloor$，其中$n$是$G$的顶点数，并且这个界是紧的。我们证明了如果$G$是一个二部图，那么$box(G) \leq \left \lceil\frac{n}{4} \right\rceil$和这个界是紧的。指出存在盒性很高但色数很低的图。例如，存在二分的（即，2可着色的）图，其盒度等于$\frac{n}{4}$。有趣的是，如果盒度非常接近$\frac{n}{2}$，那么色数也必须非常高。特别地，我们证明了，如果$box(G) = \frac{n}{2} - s$，$s \geq 0$，那么$χ(G) \geq \frac{n}{2s+2}$，其中$χ(G)$是$G$的色数。

英文摘要：

A $k$-dimensional box is the cartesian product $R_1 \times R_2 \times ... \times R_k$ where each $R_i$ is a closed interval on the real line. The {\it boxicity} of a graph $G$, denoted as $box(G)$, is the minimum integer $k$ such that $G$ is the intersection graph of a collection of $k$-dimensional boxes. A unit cube in $k$-dimensional space or a $k$-cube is defined as the cartesian product $R_1 \times R_2 \times ... \times R_k$ where each $R_i$ is a closed interval on the real line of the form $[a_i, a_{i}+1]$. The {\it cubicity} of $G$, denoted as $cub(G)$, is the minimum $k$ such that $G$ is the intersection graph of a collection of $k$-cubes. In this paper we show that $cub(G) \leq t + \left \lceil \log (n - t)\right\rceil - 1$ and $box(G) \leq \left \lfloor\frac{t}{2}\right\rfloor + 1$, where $t$ is the cardinality of the minimum vertex cover of $G$ and $n$ is the number of vertices of $G$. We also show the tightness of these upper bounds. F. S. Roberts in his pioneering paper on boxicity and cubicity had shown that for a graph $G$, $box(G) \leq \left \lfloor\frac{n}{2} \right \rfloor$, where $n$ is the number of vertices of $G$, and this bound is tight. We show that if $G$ is a bipartite graph then $box(G) \leq \left \lceil\frac{n}{4} \right\rceil$ and this bound is tight. We point out that there exist graphs of very high boxicity but with very low chromatic number. For example there exist bipartite (i.e., 2 colorable) graphs with boxicity equal to $\frac{n}{4}$. Interestingly, if boxicity is very close to $\frac{n}{2}$, then chromatic number also has to be very high. In particular, we show that if $box(G) = \frac{n}{2} - s$, $s \geq 0$, then $χ(G) \geq \frac{n}{2s+2}$, where $χ(G)$ is the chromatic number of $G$.

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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