The biclique cover number $(\text{bc})$ of a graph $G$ is referred to as the least number of complete bipartite (biclique) subgraphs that are required to cover all the edges of the graph. In this paper, we show that the biclique cover number $(\text{bc})$ of a graph $G$ is no less than $\lceil \log_2(\text{mc}(G^c)) \rceil$, where $\text{mc}(G^c)$ is the number of maximal cliques of the complementary graph $G^c$, i.e., the number of maximal independent sets of $G$. We also show that $\text{bc}(G) \leq \chi_r'(\mathcal{T}_{\mathcal{K}^c})$ where $G$ is a co-chordal graph such that each vertex is in at most two maximal independent sets and $\chi_r'(\mathcal{T}_{\mathcal{K}^c})$ is the optimal edge-ranking number of a clique tree of $G^c$. By identifying the new lower and upper bounds of $\text{bc}(G)$, we prove that $\text{bc}(G) = \lceil \log_2(\text{mc}(G^c)) \rceil$ if $G^c$ is a path or windmill graph.

翻译：图$G$的双团覆盖数$(\text{bc})$指覆盖该图所有边所需的最少完全二部（双团）子图数量。本文证明：图$G$的双团覆盖数$(\text{bc})$不小于$\lceil \log_2(\text{mc}(G^c)) \rceil$，其中$\text{mc}(G^c)$为补图$G^c$的最大团数量，即$G$的最大独立集数量。同时证明，当$G$为共弦图且每个顶点至多属于两个最大独立集时，$\text{bc}(G) \leq \chi_r'(\mathcal{T}_{\mathcal{K}^c})$成立，其中$\chi_r'(\mathcal{T}_{\mathcal{K}^c})$为$G^c$的团树的最优边排序数。通过确定$\text{bc}(G)$的新下界与上界，我们证明：若$G^c$为路径图或风车图，则$\text{bc}(G) = \lceil \log_2(\text{mc}(G^c)) \rceil$。