The problems of determining the permutation-representation number (prn) and the representation number of bipartite graphs are open in the literature. Moreover, the decision problem corresponding to the determination of the prn of a bipartite graph is NP-complete. However, these numbers were established for certain subclasses of bipartite graphs, e.g., for crown graphs. Further, it was conjectured that the crown graphs have the highest representation number among the bipartite graphs. In this work, first, we reconcile the relation between the prn of a comparability graph and the dimension of its induced poset and review the upper bounds on the prn of bipartite graphs. Then, we study the prn of bipartite graphs using the notion called neighborhood graphs. This approach substantiates the aforesaid conjecture and gives us theoretical evidence. In this connection, we devise a polynomial-time procedure to construct a word that represents a given bipartite graph permutationally. Accordingly, we provide a better upper bound for the prn of bipartite graphs. Further, we construct a class of bipartite graphs, viz., extended crown graphs, defined over posets and investigate its prn using the neighborhood graphs.

翻译：确定双部图的置换表示数（prn）和表示数的问题在文献中仍为开放问题。此外，对应于确定双部图prn的决策问题是NP完全的。然而，这些数已在某些双部图子类（如冠图）中得到确立。进一步地，曾有猜想认为冠图在双部图中具有最高的表示数。在本工作中，我们首先调和了可比图的prn与其诱导偏序集维数之间的关系，并回顾了双部图prn的上界。接着，我们利用称为邻域图的概念研究双部图的prn。该方法证实了前述猜想，并为我们提供了理论依据。在此基础上，我们设计了一个多项式时间过程来构造一个可置换表示给定双部图的单词。据此，我们给出了双部图prn的一个更优上界。进一步地，我们构造了一类定义在偏序集上的双部图（即扩展冠图），并利用邻域图研究了其prn。