arXiv 0810.26972008-10-15cs.DM

关于二部图的立方性

On the cubicity of bipartite graphs

L. Sunil Chandran, Anita Das, Naveen Sivadasan

备注：: 7 pages

摘要：

{ \it} $k$ -维中的单位立方体（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$[a_i, a_i+1]$形式的实数线上的闭区间. $G$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。许多NP-完全图问题在低立方度图中可以有效地求解或具有很好的逼近比。在大多数情况下，第一步是得到给定图的低维立方体表示。已知对于图$G$，$cub(G) \leq \lfloor\frac{2n}{3}\rfloor$ .最近的研究表明，对于图$G$，有$cub(G) \leq 4(Δ+ 1)\ln n$，其中$n$和$Δ$分别是$G$的顶点数和最大度。本文证明了：对于一个二部图$G = (A \cup B, E)$，其中$|A| = n_1$，$|B| = n_2$，$n_1 \leq n_2$，和$Δ' = \min\{Δ_A, Δ_B\}$，其中$Δ_A = {max}_{a \in A}d(a)$和$Δ_B = {max}_{b \in B}d(b)$，$d(a)$和$d(b)$分别是$a$和$b$在$G$中的度，$cub(G) \leq 2(Δ'+2) \lceil \ln n_2 \rceil$ .我们还给出了一个有效的随机算法来构造$G$在$3(Δ'+2)\lceil \ln n_2 \rceil$维上的立方体表示。读者可能会注意到，通常$Δ'$可以比$Δ$小得多。

英文摘要：

{\it A unit cube in $k$-dimension (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. Many NP-complete graph problems can be solved efficiently or have good approximation ratios in graphs of low cubicity. In most of these cases the first step is to get a low dimensional cube representation of the given graph. It is known that for a graph $G$, $cub(G) \leq \lfloor\frac{2n}{3}\rfloor$. Recently it has been shown that for a graph $G$, $cub(G) \leq 4(Δ+ 1)\ln n$, where $n$ and $Δ$ are the number of vertices and maximum degree of $G$, respectively. In this paper, we show that for a bipartite graph $G = (A \cup B, E)$ with $|A| = n_1$, $|B| = n_2$, $n_1 \leq n_2$, and $Δ' = \min\{Δ_A, Δ_B\}$, where $Δ_A = {max}_{a \in A}d(a)$ and $Δ_B = {max}_{b \in B}d(b)$, $d(a)$ and $d(b)$ being the degree of $a$ and $b$ in $G$ respectively, $cub(G) \leq 2(Δ'+2) \lceil \ln n_2 \rceil$. We also give an efficient randomized algorithm to construct the cube representation of $G$ in $3(Δ'+2)\lceil \ln n_2 \rceil$ dimensions. The reader may note that in general $Δ'$ can be much smaller than $Δ$.}

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