Count Matroids of Group-Labeled Graphs

TL;DR

This paper extends the concept of count matroids of graphs by incorporating group labelings, introducing near-balancedness to define a new class of matroids with modified independence conditions.

Contribution

It introduces a novel notion called near-balancedness and develops a new class of matroids based on group-labeled graphs, expanding the theory of count matroids.

Findings

Defined near-balancedness for group-labeled graphs

Identified a new class of matroids with modified independence conditions

Extended the theory of count matroids using group labelings

Abstract

A graph $G=(V,E)$ is called $(k,\ell)$-sparse if $|F|\leq k|V(F)|-\ell$ for any nonempty $F\subseteq E$, where $V(F)$ denotes the set of vertices incident to $F$. It is known that the family of the edge sets of $(k,\ell)$-sparse subgraphs forms the family of independent sets of a matroid, called the $(k,\ell)$-count matroid of $G$. In this paper we shall investigate lifts of the $(k,\ell)$-count matroid by using group labelings on the edge set. By introducing a new notion called near-balancedness, we shall identify a new class of matroids, where the independence condition is described as a count condition of the form $|F|\leq k|V(F)|-\ell +\alpha_{\psi}(F)$ for some function $\alpha_{\psi}$ determined by a given group labeling $\psi$ on $E$.