Exact Biclique Partition number of Split Graphs

Abstract: The biclique partition number of a graph (G), denoted ( \operatorname{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 ( \operatorname{bp}(G) = \operatorname{mc}(G^c) - 1 ), where ( \operatorname{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.