On the $H$-space of a random graph

TL;DR

This paper investigates the conditions under which the $H$-space of a random graph $G_{n,p}$ equals a naturally defined subspace, focusing on strictly 2-balanced graphs and the probability of this equality holding.

Contribution

It establishes that for strictly 2-balanced graphs, the $H$-space equals the natural subspace with high probability when every edge is contained in a copy of $H$.

Findings

Equality of $H$-space and $ ext{W}_H(G)$ holds w.h.p. under certain conditions.

Results apply specifically to strictly 2-balanced graphs.

Provides probabilistic thresholds for the $H$-space properties in random graphs.

Abstract

The edge space $\mathcal{E}(G)$ of a graph $G$ is the vector space $\mathbb{F}_2^{E(G)}$ with members naturally identified with subgraphs of $G$, and the $H$-space is the subspace $\mathcal{C}_H(G)$ of $ \mathcal{E}(G)$ spanned by copies of the graph $H$. We are interested in when the random graph $G = G_{n,p}$ is likely to satisfy \[\mathcal{C}_H(G) = \mathcal{W}_H(G),\] where $\mathcal{W}_H(G)$ takes one of four natural values, depending on the value of $\mathcal{C}_H(K_n)$. We show that for strictly $2$-balanced $H$, w.h.p. the above equality holds whenever every edge of $G$ is in a copy of $H$.