Exact Biclique Partition number of Split Graphs
Fuente:
arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866912984072192000 |
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| author | Babu, Anand Jacob, Ashwin |
| author_facet | Babu, Anand Jacob, Ashwin |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_08114 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Exact Biclique Partition number of Split Graphs Babu, Anand Jacob, Ashwin Combinatorics Discrete Mathematics 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. |
| title | Exact Biclique Partition number of Split Graphs |
| topic | Combinatorics Discrete Mathematics |
| url | https://arxiv.org/abs/2507.08114 |