Exact Biclique Partition number of Split Graphs

The biclique partition number of a graph $G$, denoted $\mathrm{bp}(G)$, is the minimum number of biclique subgraphs that partition the edge set of $G$. The Graham-Pollak theorem states that the complete graph on $n$ vertices cannot be partitioned into fewer than $n-1$ bicliques. In this note, we show that for any split graph $G$, the biclique partition number satisfies $\mathrm{bp}(G)=\mathrm{mc}(G^c)-1$, where $\mathrm{mc}(G^c)$ denotes the number of maximal cliques in the complement of $G$. This extends the celebrated Graham-Pollak theorem to a broader class of graphs.