A counterexample to the conjecture on Biclique Partition number of Split Graphs and related problems

TL;DR

This paper disproves a conjecture relating biclique partition numbers and maximal cliques in split graphs by providing a counterexample and an infinite family of counterexamples, also solving an open problem on tournaments.

Contribution

It presents the first counterexamples to the conjecture on biclique partition numbers in split graphs and explores structural properties of such partitions.

Findings

Counterexample disproves the conjecture for split graphs.

Constructs an infinite family of counterexamples.

Solves an open problem on singular tournaments with binary rank.

Abstract

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 \(n\).