Chordal bipartite graphs, biclique vertex partitions and Castelnuovo-Mumford regularity of $1$-subdivision graphs
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arXiv
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| Autori principali: | , , , |
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| Natura: | Preprint |
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2024
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| _version_ | 1866914980727619584 |
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| author | Civan, Yusuf Deniz, Zakir Duginov, Oleg Yetim, Mehmet Akif |
| author_facet | Civan, Yusuf Deniz, Zakir Duginov, Oleg Yetim, Mehmet Akif |
| contents | A biclique in a graph $G$ is a complete bipartite subgraph (not necessarily induced), and the least positive integer $k$ for which the vertex set of $G$ can be partitioned into at most $k$ bicliques is the biclique vertex partition number $bp(G)$ of $G$. We prove that the inequality $reg(S(G))\geq |G|-bp(G)$ holds for every graph $G$, where $S(G)$ is the $1$-subdivision graph of $G$ and $reg(S(G))$ denotes the (Castelnuovo-Mumford) regularity of the graph $S(G)$. In particular, we show that the equality $reg(S(B))=|B|-bp(B)$ holds provided that $B$ is a chordal bipartite graph. Furthermore, for every chordal bipartite graph $B$, we prove that the independence complex of $S(B)$ is either contractible or homotopy equivalent to a sphere, and provide a polynomial time checkable criteria for when it is contractible, and describe the dimension of the sphere when it is not. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_15213 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Chordal bipartite graphs, biclique vertex partitions and Castelnuovo-Mumford regularity of $1$-subdivision graphs Civan, Yusuf Deniz, Zakir Duginov, Oleg Yetim, Mehmet Akif Combinatorics 13F55, 05E40, 05E45, 05C69 A biclique in a graph $G$ is a complete bipartite subgraph (not necessarily induced), and the least positive integer $k$ for which the vertex set of $G$ can be partitioned into at most $k$ bicliques is the biclique vertex partition number $bp(G)$ of $G$. We prove that the inequality $reg(S(G))\geq |G|-bp(G)$ holds for every graph $G$, where $S(G)$ is the $1$-subdivision graph of $G$ and $reg(S(G))$ denotes the (Castelnuovo-Mumford) regularity of the graph $S(G)$. In particular, we show that the equality $reg(S(B))=|B|-bp(B)$ holds provided that $B$ is a chordal bipartite graph. Furthermore, for every chordal bipartite graph $B$, we prove that the independence complex of $S(B)$ is either contractible or homotopy equivalent to a sphere, and provide a polynomial time checkable criteria for when it is contractible, and describe the dimension of the sphere when it is not. |
| title | Chordal bipartite graphs, biclique vertex partitions and Castelnuovo-Mumford regularity of $1$-subdivision graphs |
| topic | Combinatorics 13F55, 05E40, 05E45, 05C69 |
| url | https://arxiv.org/abs/2410.15213 |