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

TL;DR

This paper characterizes bipartite graphs where the induced matching number equals the ordered matching number, and explores algebraic properties of their edge and cover ideals, providing formulas for specific subgraph counts.

Contribution

It characterizes bipartite graphs with equal induced and ordered matching numbers and analyzes algebraic invariants of their associated ideals.

Findings

Characterization of bipartite graphs with indm(G) = ordm(G)

Formulas for counting certain spanning subgraphs of K_{m,n}

Explicit counts for m=2,3,4 when m <= n

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.