Chordal bipartite graphs, biclique vertex partitions and Castelnuovo-Mumford regularity of $1$-subdivision graphs

Abstract

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.

Figures (8)

• Figure 1: The graph $R^4_4$.
• Figure 2: A chordal bipartite graph $B$ such that $\gamma^i(B)=2< 3=\operatorname{bp}(B)$.
• Figure 3: The $\mathcal{C}\mathcal{S}\mathcal{B}\mathcal{E}$-graph $C_6(4)$.
• Figure 4: Two possible cases of Claim $1$.
• Figure 5: The chordal bipartite graph $B_4$.

Theorems & Definitions (47)

• Theorem 1
• Corollary 2
• Theorem 3
• Theorem 4
• Remark 5
• Proposition 6
• Definition 8
• Corollary 9
• Lemma 10
• Lemma 11