Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: On the Cycle Space of a Random Graph   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).

4
The categories of this paper are: cond-mat
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The categories of this paper are: quant-ph
0
The categories of this paper are: hep-ph
3
The categories of this paper are: physics
36
The categories of this paper are: cmp-lg
35
The categories of this paper are: mtrl-th
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The categories of this paper are: funct-an
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The categories of this paper are: nucl-th
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The categories of this paper are: patt-sol
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The categories of this paper are: plasm-ph
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The categories of this paper are: q-fin
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The categories of this paper are: solv-int
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The categories of this paper are: eess
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The categories of this paper are: econ
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The categories of this paper are: supr-con
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The categories of this paper are: hep-ex