星状三自由图的盒度和立方度
Boxicity and Cubicity of Asteroidal Triple free graphs
摘要：
轴平行$d$维盒是笛卡尔积$R_1 \times R_2 \times ... \times R_d$，其中每个$R_i$是实数线上的闭合区间。图$G$的{ \it boxicity}表示为$\boxi(G)$，是最小整数$d$，使得$G$可以表示为$d$维盒的集合的交集图。$d$维空间中的轴平行单位立方体或$d$立方体被定义为笛卡尔积$R_1 \times R_2 \times ... \times R_d$，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立方度}表示为$\cub(G)$，是最小整数$d$，使得$G$可以表示为$d$立方的集合的交集图。令$S(m)$表示$m+1$节点上的星形图。定义图$G$的{ \it爪数}为使$S(k)$是$G$的导出子图的最大正整数$k$，记为$\claw$ .设$G$是一个无AT图，其色数为$χ(G)$，爪数为$\claw$ .本文将证明$\boxi(G) \leq χ(G)$和这个界是紧的.我们还表明，$\cub(G) \leq \boxi(G)(\ceil{\log_2 \claw} +2)$ $\leq$ $χ(G)(\ceil{\log_2 \claw} +2)$。如果$G$是围长至少为5的无AT图，则$\boxi(G) \leq 2$和$\cub(G) \leq 2\ceil{\log_2 \claw} +4$。
英文摘要：
An axis parallel $d$-dimensional box is the Cartesian product $R_1 \times R_2 \times ... \times R_d$ where each $R_i$ is a closed interval on the real line. The {\it boxicity} of a graph $G$, denoted as $\boxi(G)$, is the minimum integer $d$ such that $G$ can be represented as the intersection graph of a collection of $d$-dimensional boxes. An axis parallel unit cube in $d$-dimensional space or a $d$-cube is defined as the Cartesian product $R_1 \times R_2 \times ... \times R_d$ 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 integer $d$ such that $G$ can be represented as the intersection graph of a collection of $d$-cubes. Let $S(m)$ denote a star graph on $m+1$ nodes. We define {\it claw number} of a graph $G$ as the largest positive integer $k$ such that $S(k)$ is an induced subgraph of $G$ and denote it as $\claw$. Let $G$ be an AT-free graph with chromatic number $χ(G)$ and claw number $\claw$. In this paper we will show that $\boxi(G) \leq χ(G)$ and this bound is tight. We also show that $\cub(G) \leq \boxi(G)(\ceil{\log_2 \claw} +2)$ $\leq$ $χ(G)(\ceil{\log_2 \claw} +2)$. If $G$ is an AT-free graph having girth at least 5 then $\boxi(G) \leq 2$ and therefore $\cub(G) \leq 2\ceil{\log_2 \claw} +4$.
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