Large Simple d-Cycles in Simplicial Complexes

TL;DR

This paper generalizes a classical graph theory result to higher-dimensional simplicial complexes, establishing a lower bound on the size of the largest simple d-cycle based on the complex's density, using matroid theory methods.

Contribution

It extends Erdős-Gallai's theorem from graphs to simplicial complexes, introducing a new bound for simple d-cycles using matroid theory.

Findings

Largest simple d-cycle size is at least the square root of the complex's density.

Generalizes classical graph results to higher dimensions.

Employs matroid theory in the context of simplicial complexes.

Abstract

We show that the size of the largest simple d-cycle in a simplicial d-complex $K$ is at least a square root of $K$'s density. This generalizes a well-known classical result of Erd\H{o}s and Gallai \cite{EG59} for graphs. We use methods from matroid theory applied to combinatorial simplicial complexes.