区间图的立方性与爪数
Cubicity of interval graphs and the claw number
摘要：
设$G(V,E)$是一个简单的无向图，其中$V$是顶点集，$E$是边集。$b$维立方体是笛卡尔积$I_1\times I_2\times...\times I_b$，其中每个$I_i$是实数线上单位长度的闭区间. $G$的\emph{cubicity}，用$\cub(G)$表示，是最小正整数$b$，使得$G$中的顶点可以以这样的方式映射到轴平行的$b$维立方体，即当且仅当它们所分配的立方体相交时，两个顶点在$G$中相邻。假设$S(m)$表示$m+1$节点上的星图。定义图的\emph{claw number} $ψ(G)$为最大正整数$m$，使得$S(m)$是$G$的导出子图.可以很容易地证明任何图的立方度至少是$\ceil{\log_2ψ(G)}$。在本文中，我们证明了，对于一个区间图$G$ $\ceil{\log_2ψ(G)}\le\cub(G)\le\ceil{\log_2ψ(G)}+2$。到目前为止，我们无法找到任何区间图与$\cub(G)>\ceil{\log_2ψ(G)}$。我们还证明了，对于一个区间图$G$，$\cub(G)\le\ceil{\log_2α}$，其中$α$是$G$的独立数。因此，在$ψ(G)=α$的特殊情况下，$\cub(G)$就是$\ceil{\log_2α}$。立方体的概念可以通过考虑盒子而不是立方体来推广。$b$维盒是笛卡尔积$I_1\times I_2\times...\times I_b$，其中每个$I_i$是实直线上的闭区间.一个图的\emph{boxicity}，记为$ box(G)$，是最小的$k$，使得$G$是$k$维盒的交图。很明显，$ box(G)\le\cub(G)$。从上面的结果可以得出，对于任何图$G$，$\cub(G)\le box(G)\ceil{\log_2α}$。
英文摘要：
Let $G(V,E)$ be a simple, undirected graph where $V$ is the set of vertices and $E$ is the set of edges. A $b$-dimensional cube is a Cartesian product $I_1\times I_2\times...\times I_b$, where each $I_i$ is a closed interval of unit length on the real line. The \emph{cubicity} of $G$, denoted by $\cub(G)$ is the minimum positive integer $b$ such that the vertices in $G$ can be mapped to axis parallel $b$-dimensional cubes in such a way that two vertices are adjacent in $G$ if and only if their assigned cubes intersect. Suppose $S(m)$ denotes a star graph on $m+1$ nodes. We define \emph{claw number} $ψ(G)$ of the graph to be the largest positive integer $m$ such that $S(m)$ is an induced subgraph of $G$. It can be easily shown that the cubicity of any graph is at least $\ceil{\log_2ψ(G)}$. In this paper, we show that, for an interval graph $G$ $\ceil{\log_2ψ(G)}\le\cub(G)\le\ceil{\log_2ψ(G)}+2$. Till now we are unable to find any interval graph with $\cub(G)>\ceil{\log_2ψ(G)}$. We also show that, for an interval graph $G$, $\cub(G)\le\ceil{\log_2α}$, where $α$ is the independence number of $G$. Therefore, in the special case of $ψ(G)=α$, $\cub(G)$ is exactly $\ceil{\log_2α}$. The concept of cubicity can be generalized by considering boxes instead of cubes. A $b$-dimensional box is a Cartesian product $I_1\times I_2\times...\times I_b$, where each $I_i$ is a closed interval on the real line. The \emph{boxicity} of a graph, denoted $ box(G)$, is the minimum $k$ such that $G$ is the intersection graph of $k$-dimensional boxes. It is clear that $ box(G)\le\cub(G)$. From the above result, it follows that for any graph $G$, $\cub(G)\le box(G)\ceil{\log_2α}$.
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