Count and cofactor matroids of highly connected graphs

TL;DR

This paper investigates the properties of count and cofactor matroids on highly connected graphs, establishing bounds that link connectivity with matroid rank and uniqueness of graph determination, extending classical theorems.

Contribution

It provides tight bounds connecting high connectivity with matroid properties, unifies previous results, and generalizes Whitney's theorem to broader classes of matroids.

Findings

Highly connected graphs have maximum rank in associated matroids after edge removal.

High connectivity ensures the matroids are connected and have high vertical connectivity.

Highly connected graphs are uniquely determined by their count and cofactor matroids.

Abstract

We consider two types of matroids defined on the edge set of a graph $G$: count matroids ${\cal M}_{k,\ell}(G)$, in which independence is defined by a sparsity count involving the parameters $k$ and $\ell$, and the (three-dimensional generic) cofactor matroid $\mathcal{C}(G)$, in which independence is defined by linear independence in the cofactor matrix of $G$. We give tight lower bounds, for each pair $(k,\ell)$, that show that if $G$ is sufficiently highly connected, then $G-e$ has maximum rank for all $e\in E(G)$, and ${\cal M}_{k,\ell}(G)$ is connected. These bounds unify and extend several previous results, including theorems of Nash-Williams and Tutte ($k=\ell$), and Lov\'asz and Yemini ($k=2, \ell=3$). We also prove that if $G$ is highly connected, then the vertical connectivity of $\mathcal{C}(G)$ is also high. We use these results to generalize Whitney's celebrated result on the graphic matroid of $G$ (which corresponds to ${\cal M}_{1,1}(G)$) to all count matroids and to the three-dimensional cofactor matroid: if $G$ is highly connected, depending on $k$ and $\ell$, then the count matroid ${\cal M}_{k,\ell}(G)$ uniquely determines $G$; and similarly, if $G$ is $14$-connected, then its cofactor matroid $\mathcal{C}(G)$ uniquely determines $G$. We also derive similar results for the $t$-fold union of the three-dimensional cofactor matroid, and use them to prove that every $24$-connected graph has a spanning tree $T$ for which $G-E(T)$ is $3$-connected, which verifies a case of a conjecture of Kriesell.