Skew circuits and circumference in a binary matroid

TL;DR

This paper investigates properties of skew circuits in binary matroids, establishing bounds on their sizes relative to the matroid's circumference and certain subset conditions.

Contribution

It introduces new bounds relating the sizes of skew circuits and the circumference in binary matroids, expanding understanding of their structural properties.

Findings

Bounds on the sum of sizes of skew circuits in terms of circumference

Existence of a constant a_k for given parameters

Structural constraints on binary matroids

Abstract

Let C_1 and C_2 be skew circuits in a binary matroid having circumference c. For any positive integer k there is a constant a_k such that if min { |A| ; C_1 \subset A \subset E-A} > a_k, then |C_1| + |C_2| < 2c -k.