Biclique Covers and Partitions
Biclique Covers and Partitions
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Biclique 盖板和隔板
DOI：
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发表时间：
2013
影响因子：
0.7
通讯作者：
Trevor Pinto
中科院分区：
文献类型：
--
作者：
Trevor Pinto
图$ g $，$ mathrm {bc}（g $）的biclique封面编号（分别biclique分区编号）（$ $ mathrm {bp}（g）$），是bicliques的最小数量 - 完整的biptite子图 - 需要覆盖（分区）$ g $的边缘。 $ mathrm {lbp}（g）$），是$ r $，因此有一个封面（分区）$ g $的边缘，由bicliques by bicliques中没有顶点，我们表明$ mathrm {bp}（g）$可以按$ MATHRM {bc}限制（g）$，尤其是$ mathrm {bp}（g）leq frac {1} {2}（3^mathrm {bc（g）} - 1）$。确实，在我们的主要结果中，我们证明$ MATHRM {lbp}（g）$也可以很大$ mathrm {lbc}（g）= 2 $，我们尝试绑定有关$ g $的其他信息答案并留下与此相关的问题。一个子立方体相交图每个子立方体都有尺寸$ r $。 。
The biclique cover number (resp. biclique partition number ) of a graph $G$, $mathrm{bc}(G$) (resp. $mathrm{bp}(G)$), is the least number of bicliques - complete bipartite subgraphs - that are needed to cover (resp. partition) the edges of $G$. The local biclique cover number (resp. local biclique partition number ) of a graph $G$, $mathrm{lbc}(G$) (resp. $mathrm{lbp}(G)$), is the least $r$ such that there is a cover (resp. partition) of the edges of $G$ by bicliques with no vertex in more than $r$ of these bicliques. We show that $mathrm{bp}(G)$ may be bounded in terms of $mathrm{bc}(G)$, in particular, $mathrm{bp}(G)leq frac{1}{2}(3^mathrm{bc(G)}-1)$. However, the analogous result does not hold for the local measures. Indeed, in our main result, we show that $mathrm{lbp}(G)$ can be arbitrarily large, even for graphs with $mathrm{lbc}(G)=2$. For such graphs, $G$, we try to bound $mathrm{lbp}(G)$ in terms of additional information about biclique covers of $G$. We both answer and leave open questions related to this. There is a well known link between biclique covers and subcube intersection graphs. We consider the problem of finding the least $r(n)$ for which every graph on $n$ vertices can be represented as a subcube intersection graph in which every subcube has dimension $r$. We reduce this problem to the much studied question of finding the least $d(n)$ such that every graph on $n$ vertices is the intersection graph of subcubes of a $d$-dimensional cube.