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Induced matching, ordered matching and Castelnuovo-Mumford regularity of bipartite graphs

A. V. Jayanthan, S. A. Seyed Fakhari, I. Swanson, S. Yassemi

TL;DR

The paper Characterizes bipartite graphs for which the induced matching number and the ordered matching number coincide, and derives algebraic consequences for edge and cover ideals in this class, notably the Castelnuovo-Mumford regularity of powers and the depth of powers of cover ideals. It develops a combinatorial framework based on $k$-sequences and $J$-sets to classify and count such graphs, and proves that for these graphs the regularity of edge-ideal powers can be explicitly bounded by the shared matching number. A central contribution is a complete classification of bipartite graphs with $\operatorname{indm}(G)=\operatorname{ordm}(G)$, plus detailed counting formulas for connected spanning subgraphs of complete bipartite graphs with $\operatorname{indm}(G)=\operatorname{ordm}(G)=2$, including closed forms for small partitions. These results bridge combinatorial graph structure and algebraic invariants, enabling precise enumeration and deeper understanding of how matching properties control algebraic complexity.

Abstract

Let G be a finite simple graph and let indm(G) and ordm(G) denote the induced matching number and the ordered matching number of G, respectively. We characterize all bipartite graphs G with indm(G) = ordm(G). We establish the Castelnuovo-Mumford regularity of powers of edge ideals and depth of powers of cover ideals for such graphs. We also give formulas for the count of connected non-isomorphic spanning subgraphs of the complete bipartite graph K_{m,n} for which indm(G) = ordm(G) = 2, with an explicit expression for the count when m = 2,3,4 and m <= n.

Induced matching, ordered matching and Castelnuovo-Mumford regularity of bipartite graphs

TL;DR

The paper Characterizes bipartite graphs for which the induced matching number and the ordered matching number coincide, and derives algebraic consequences for edge and cover ideals in this class, notably the Castelnuovo-Mumford regularity of powers and the depth of powers of cover ideals. It develops a combinatorial framework based on -sequences and -sets to classify and count such graphs, and proves that for these graphs the regularity of edge-ideal powers can be explicitly bounded by the shared matching number. A central contribution is a complete classification of bipartite graphs with , plus detailed counting formulas for connected spanning subgraphs of complete bipartite graphs with , including closed forms for small partitions. These results bridge combinatorial graph structure and algebraic invariants, enabling precise enumeration and deeper understanding of how matching properties control algebraic complexity.

Abstract

Let G be a finite simple graph and let indm(G) and ordm(G) denote the induced matching number and the ordered matching number of G, respectively. We characterize all bipartite graphs G with indm(G) = ordm(G). We establish the Castelnuovo-Mumford regularity of powers of edge ideals and depth of powers of cover ideals for such graphs. We also give formulas for the count of connected non-isomorphic spanning subgraphs of the complete bipartite graph K_{m,n} for which indm(G) = ordm(G) = 2, with an explicit expression for the count when m = 2,3,4 and m <= n.
Paper Structure (6 sections, 18 theorems, 48 equations, 1 figure)

This paper contains 6 sections, 18 theorems, 48 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Equality of induced and ordered matching numbers
  4. Set-up of $k$-sequences and J-sets
  5. Bipartite graphs with induced and ordered matching $2$
  6. Appendix: Some explicit summations

Key Result

theorem 1

Assume that $G$ is a bipartite graph with $\operatorname{ind-match}(G)=\operatorname{ord-match}(G)$ and suppose $d=|V(G)|$. Let $J(G)$ denote the cover ideal of $G$. Then for every integer $k\geq 1$, we have

Figures (1)

  • Figure 1: $G_{2,4}$

Theorems & Definitions (30)

  • theorem 1
  • proof
  • theorem 2
  • proof
  • Example 1
  • Proposition 2
  • proof
  • theorem 3
  • Definition 3
  • Proposition 4
  • ...and 20 more