Exact Biclique Partition number of Split Graphs

TL;DR

This paper extends the Graham-Pollak theorem by establishing an exact formula for the biclique partition number of split graphs, linking it to the number of maximal cliques in their complements.

Contribution

It provides a precise characterization of the biclique partition number for split graphs, generalizing a classical theorem to a new class of graphs.

Findings

For split graphs, bp(G) = mc(G^c) - 1.

Extends Graham-Pollak theorem to split graphs.

Provides an exact formula for biclique partition number.

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.