On the Cycle Space of a Random Graph

TL;DR

This paper investigates the cycle space of random graphs, showing that for odd cycles of length at least 3, the subspace spanned by these cycles almost surely covers the entire cycle space as the graph grows, extending previous results for triangles.

Contribution

It generalizes the understanding of cycle spaces in random graphs to all odd cycle lengths, demonstrating that the cycles of length fectively span the entire cycle space under certain conditions.

Findings

For odd fectively span the entire cycle space.

The probability that cycles of length fectively cover the cycle space approaches 1.

Extends prior results from triangles to all odd cycles.

Abstract

Write $\mathcal{C}(G)$ for the cycle space of a graph $G$, $\mathcal{C}_\kappa(G)$ for the subspace of $\mathcal{C}(G)$ spanned by the copies of the $\kappa$-cycle $C_\kappa$ in $G$, $\mathcal{T}_\kappa$ for the class of graphs satisfying $\mathcal{C}_\kappa(G)=\mathcal{C}(G)$, and $\mathcal{Q}_\kappa$ for the class of graphs each of whose edges lies in a $C_\kappa$. We prove that for every odd $\kappa \geq 3$ and $G=G_{n,p}$, \[\max_p \, \Pr(G \in \mathcal{Q}_\kappa \setminus \mathcal{T}_\kappa) \rightarrow 0;\] so the $C_\kappa$'s of a random graph span its cycle space as soon as they cover its edges. For $\kappa=3$ this was shown by DeMarco, Hamm and Kahn (2013).