The biclique partition number of a graph \(G\), denoted \( \operatorname{bp}(G)\), is the minimum number of biclique subgraphs needed to partition the edge set of $G$. Lyu and Hicks \cite{lyu2023finding} posed the open problem of whether \( \operatorname{bp}(G) = \operatorname{mc}(G^c) - 1 \) holds for every co-chordal graph or split graph, where \( \operatorname{mc}(G^c) \) denotes the number of maximal cliques in the complement of \( G \). Such a result would extend the celebrated Graham--Pollak theorem to a more general class of graphs. In this note, we answer this problem in the negative by providing a counterexample using a split graph. We also construct an infinite family of counterexamples and prove some structural properties of biclique partitions of split graphs. Finally, we solve an open problem posed by Siewert \cite{siewert2000biclique} on the existence of singular \(n\)-tournaments with binary rank \(n\).

翻译：图\(G\)的二部团划分数，记作\(\operatorname{bp}(G)\)，是指划分图\(G\)的边集所需的最小二部子图数量。Lyu和Hicks \cite{lyu2023finding} 提出了一个开放问题：是否对于每个共弦图或分裂图，都有\(\operatorname{bp}(G) = \operatorname{mc}(G^c) - 1\)成立，其中\(\operatorname{mc}(G^c)\)表示补图\(G^c\)中最大团的数量。这一结果本可将著名的Graham--Pollak定理推广到更一般的图类。在本篇短文中，我们通过构造一个分裂图作为反例，对此问题给出了否定回答。我们还构造了一个无限反例族，并证明了分裂图二部团划分的一些结构性质。最后，我们解决了Siewert \cite{siewert2000biclique} 提出的关于二元秩为\(n\)的奇异\(n\)-锦标赛存在性的一个开放问题。