Any finite simple graph $G = (V,E)$ can be represented by a collection $\mathscr{C}$ of subsets of $V$ such that $uv\in E$ if and only if $u$ and $v$ appear together in an odd number of sets in $\mathscr{C}$. Let $c_2(G)$ denote the minimum cardinality of such a collection. This invariant is equivalent to the minimum dimension of a faithful orthogonal representation of $G$ over $\mathbb{F}_2$ and is closely connected to the minimum rank of $G$. We show that $c_2 (G) = \operatorname{mr}(G,\mathbb{F}_2)$ when $\operatorname{mr}(G,\mathbb{F}_2)$ is odd, or when $G$ is a forest. Otherwise, $\operatorname{mr}(G,\mathbb{F}_2)\leqslant c_2 (G)\leqslant \operatorname{mr}(G,\mathbb{F}_2)+1$. Furthermore, we show that the following are equivalent for any graph $G$ with at least one edge: i. $c_2(G)=\operatorname{mr}(G,\mathbb{F}_2)+1$; ii. the adjacency matrix of $G$ is the unique matrix of rank $\operatorname{mr}(G,\mathbb{F}_2)$ which fits $G$ over $\mathbb{F}_2$; iii. there is a minimum collection $\mathscr{C}$ as described in which every vertex appears an even number of times; and iv. for every component $G'$ of $G$, $c_2(G') = \operatorname{mr}(G',\mathbb{F}_2) + 1$. We also show that, for these graphs, $\operatorname{mr}(G,\mathbb{F}_2)$ is twice the minimum number of tricliques whose symmetric difference of edge sets is $E$. Additionally, we provide a set of upper bounds on $c_2(G)$ in terms of the order, size, and vertex cover number of $G$. Finally, we show that the class of graphs with $c_2(G)\leqslant k$ is hereditary and finitely defined. For odd $k$, the sets of minimal forbidden induced subgraphs are the same as those for the property $\operatorname{mr}(G,\mathbb{F}_2)\leq k$, and we exhibit this set for $c_2(G) \leqslant 2$.

Given a finite simple graph G, an odd cover of G is a collection of complete bipartite graphs, or bicliques, in which each edge of G appears in an odd number of bicliques and each non-edge of G appears in an even number of bicliques. We denote the minimum cardinality of an odd cover of G by b2(G) and prove that b2(G) is bounded below by half of the rank over F2 of the adjacency matrix of G. We show that this lower bound is tight in the case when G is a bipartite graph and almost tight when G is an odd cycle. However, we also present an infinite family of graphs which shows that this lower bound can be arbitrarily far away from b2(G).Babai and Frankl (1992) proposed the "odd cover problem," which in our language is equivalent to determining b2(Kn). Radhakrishnan, Sen, and Vishwanathan (2000) determined b2(Kn) for an infinite but density zero subset of positive integers n. In this paper, we determine b2(Kn) for a density 3/8 subset of the positive integers.

Given a finite simple graph , an odd cover of is a collection of complete bipartite graphs, or bicliques, in which each edge of appears in an odd number of bicliques, and each nonedge of appears in an even number of bicliques. We denote the minimum cardinality of an odd cover of by and prove that is bounded below by half of the rank over of the adjacency matrix of . We show that this lower bound is tight in the case when is a bipartite graph and almost tight when is an odd cycle. However, we also present an infinite family of graphs which shows that this lower bound can be arbitrarily far away from . Babai and Frankl proposed the “odd cover problem,” which in our language is equivalent to determining . In this paper, we determine that is when and is when is equivalent to 1 or modulo 8.

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