Fundamental cycles in grid graphs

TL;DR

This paper proves that the average length of fundamental cycles in an n-by-n grid graph grows at least logarithmically with n, confirming a conjecture related to binary matroids.

Contribution

It establishes a tight lower bound on the average length of fundamental cycles in grid graphs, answering a question about sparse binary matroid representations.

Findings

Average fundamental cycle length is at least logarithmic in grid size

Bound is asymptotically tight

Answers a question posed by McCarty regarding binary matroids

Abstract

We show that the average length of a fundamental cycle with respect to any fixed spanning tree of the $n\times n$ square grid is at least $\Omega(\log n)$; the bound is asymptotically tight. This result answers in the affirmative a question posed by McCarty in relation to sparse representations of binary matroids.