A \textit{biclique} is a maximal induced complete bipartite subgraph. The \textit{biclique graph} of a graph $H$, denoted by $KB(H)$, is the intersection graph of the family of all bicliques of $H$. In this work we address the following question: Given a biclique graph $G=KB(H)$, is it possible to remove a vertex $q$ of $G$, such that $G - \{q\}$ is a biclique graph? And if possible, can we obtain a graph $H'$ such that $G - \{q\} = KB(H')$? We show that the general question has a "no" for answer. However, we prove that if $G$ has a vertex $q$ such that $d(q) = 2$, then $G-\{q\}$ is a biclique graph and we show how to obtain $H'$.
Biclique immersions in graphs with independence number 2
Universidade de São Paulo（圣保罗大学）；Universidad de Valparaíso（瓦尔帕莱索大学）；Universidade Federal do ABC（ABC联邦大学）；CONICET（阿根廷国家科学研究委员会）；Universidad Nacional de San Luis（圣路易斯国立大学）
On bipartization of cubic graphs by removal of an independent set
本文研究了三次图的一个新问题：通过删除足够大的独立集$I$来实现三次图$Q$的二分化。它可以表示如下：\emph{Given a connected $n$-vertex tripartite cubic graph $Q=(V,E)$ with independence number $α(Q)$, does $Q$ contain an independent set $I$ of size $k$ such that $Q-I$ is bipartite?}我们感兴趣的是，$k$的值对这个问题的答案是肯定的。我们建设性地证明，如果$α(Q) \geq 4n/10$，那么答案是肯定的，每个$k$履行$\lfloor (n-α(Q))/2 \rfloor \leq k \leq α(Q)$。对于具有$α(Q)<4n/10$的三次图，是否可以构造类似的图仍然是一个悬而未决的问题。接下来，我们证明了这个问题与$α(Q)\geq 4n/10$和$k$满足不等式$\lfloor n/3 \rfloor \leq k \leq α(Q)$可以与半公平图3-着色，其中一个颜色类的大小为$k$，其余顶点诱导的子图是公平2-着色。这意味着$Q$具有$(k, \lceil(n-k)/2\rceil, \lfloor (n-k)/2 \rfloor)$类型的着色。
