Number of edge-disjoint cycles in a holey graph 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 $\Gamma$?
 A: For a graph $G$, let $\nu(G)$ be the maximum number of edge-disjoint cycles, and let $\tau(G)$ be the minimum size of a set of edges $X$ such that $G-X$ has no cycles.  Note that for a connected graph $G$ with $H^1(G) \cong \mathbb{Z}^d$, we have $d = \tau(G) \geq \nu(G)$, so $\nu(G) \gg d$ is impossible (although I assume you were hoping for $\nu(G) \in \Omega(d)$).  This is also impossible, but we can get close.
Theorem. There exists a function $f(k)=O(k \log k)$ such that for every graph $G$,  $\tau(G) \leq f(\nu(G))$.
This is actually an exercise in Diestel's graph theory textbook.  In other words, $\nu(\Gamma) \geq f^{-1}(d)$.  As noted by  Gjergji Zaimi in a comment to the other answer, this bound is actually best possible (up to a constant factor) due to a classic example of Erdős and Pósa.  Indeed there is huge body of related work, which are all called "Erdős-Pósa theorems."  See this survey paper of Raymond and Thilikos or this webpage for more information.
A: The paper
Harant, Jochen; Rautenbach, Dieter; Recht, Peter; Regen, Friedrich. Packing edge-disjoint cycles in graphs and the cyclomatic number. Discrete Math. 310 (2010), no. 9, 1456--1462
constructs graphs for which the difference between $d$ and the nmaximum number of edge-disjoint cycles is $k$ for any $k \in \mathbb{N}$. If I understand the question correctly this shows you can't have a bound desired without some restrictions on the graphs (I have not thought more about the graphs described in the comments).
Edit: I posted in a rush. This paper and it's references are relevant, but does not answer the question. See comments below.
