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Let $\Gamma$ be a connected graph with $H^1(\Gamma) \sim \mathbb{Z}^d$. Can we give a lower bound (preferably of the form $\gg d$) on the maximal number of edge-disjoint cycles one can find in $\Gamma$?

Let $\Gamma$ be a connected graph with $H^1(\Gamma) \cong \mathbb{Z}^d$. Can we give a lower bound (preferably of the form $\gg d$) on the maximal number of edge-disjoint cycles one can find in $\Gamma$?