[Paper Review] Projective dimension of (hyper)graphs and the Castelnuovo-Mumford regularity of bipartite graphs

Türker Bı́yı́koğlu, Yusuf Civan|arXiv (Cornell University)|May 10, 2016

Commutative Algebra and Its Applications19 references3 citations

TL;DR

This paper establishes a fundamental inequality linking the projective dimension of a (hyper)graph to the Castelnuovo-Mumford regularity of its Levi graph, proving that proj-dim(ℋ) ≤ reg(ℒ(ℋ)) for any hypergraph ℋ. It further shows that for any graph G, there exists an induced subgraph H such that proj-dim(G) = reg(S(H)), where S(H) is the subdivision graph of H, and constructs explicit families of graphs demonstrating that many known bounds on projective dimension are far from tight.

ABSTRACT

We prove that the projective dimension of any (hyper)graph can be bounded from above by the (Castelnuovo-Mumford) regularity of its Levi graph (or incidence bipartite graph). This in particular brings the use of regularity's upper bounds on the calculation of projective dimension of (hyper)graphs. When G is just a (simple) graph, we prove that there exists an induced subgraph H of G such that prod-dim(G)=reg(S(H)), where S(H) is the subdivision graph of H. Moreover, we show that known upper bounds on prod-dim(G) involving domination parameters are in fact upper bounds to reg(S(G)).

Motivation & Objective

• To explore the combinatorial nature of Terai's duality between projective dimension and regularity in monomial ideals.
• To establish a general inequality relating the projective dimension of a (hyper)graph to the regularity of its Levi graph.
• To demonstrate that known upper bounds on projective dimension involving domination parameters are actually bounds on the regularity of subdivision graphs.
• To construct explicit graph families showing that existing bounds on projective dimension are not tight.

Proposed method

• Use of Terai's duality between projective dimension and regularity of Alexander duals to relate combinatorial invariants.
• Definition and analysis of the Levi graph ℒ(ℋ) as the incidence bipartite graph of a hypergraph ℋ.
• Introduction of the subdivision graph S(H) of a graph H and its role in relating proj-dim(H) to reg(S(H)).
• Definition of projectively prime graphs over a field ℬk, where removing any vertex strictly decreases the projective dimension.
• Application of the inequality proj-dim(G) ≤ max{proj-dim(G−N[x])+deg(x), proj-dim(G−x)+1} to construct extremal examples.
• Construction of graph families G_a and H_b using complete bipartite graphs and path components to demonstrate looseness of bounds.

Experimental results

Research questions

• RQ1 Does the inequality proj-dim(ℋ) ≤ reg(ℒ(ℋ)) hold for all (hyper)graphs ℋ, and what does it imply for projective dimension calculations?
• RQ2 Can the projective dimension of a graph G be realized as the regularity of the subdivision graph of some induced subgraph H of G?
• RQ3 How do domination parameters like independence number i(G), domination number γ(G), and edge domination number ε(G) relate to the regularity of subdivision graphs and projective dimension?
• RQ4 Are known upper bounds on proj-dim(G) involving domination parameters tight, or can they be significantly improved?
• RQ5 What structural properties characterize projectively prime graphs, and how can they be used to construct extremal examples?

Key findings

• For any hypergraph ℋ, proj-dim(ℋ) ≤ reg(ℒ(ℋ)) holds, where ℒ(ℋ) is the Levi graph of ℋ, enabling use of regularity bounds to estimate projective dimension.
• For any graph G, there exists an induced subgraph H such that proj-dim(G) = reg(S(H)), where S(H) is the subdivision graph of H.
• Known upper bounds on proj-dim(G) involving domination parameters (e.g., |G|−γ(G)) are actually upper bounds on reg(S(G)), not necessarily on proj-dim(G).
• For each a ≥ 1, there exists a graph G_a such that |G_a|−i(G_a)+a < proj-dim(G_a) < |G_a|−γ(G_a)−a, showing that both bounds are far from tight.
• For each b ≥ 1, there exists a graph H_b such that |H_b|−γ(H_b)+b < proj-dim(H_b) < |H_b|−ε(H_b)−b, further confirming looseness of domination-based bounds.
• The construction of projectively prime graphs allows for the creation of extremal examples where known bounds on projective dimension are not sharp.