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

TL;DR

This paper investigates the Castelnuovo-Mumford regularity of 1-subdivision graphs, establishing bounds related to biclique partitions and characterizing the topological structure of their independence complexes.

Contribution

It proves a lower bound on regularity in terms of biclique partition number and characterizes when equality holds for chordal bipartite graphs, also analyzing their independence complexes.

Findings

Proves that reg(S(G)) ≥ |G| - bp(G) for all graphs G.

Shows equality reg(S(B)) = |B| - bp(B) for chordal bipartite graphs B.

Classifies the homotopy type of the independence complex of S(B) as either contractible or a sphere.

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.