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Hi, I'm very new to graph theory and have a question.

For each positive integer $k$, what is the largest number of bridges in any $k$-vertex graph?


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Welcome to math.SE: since you are new, I wanted to let you know a few things about the site. In order to get the best possible answers, it is helpful if you say in what context you encountered the problem, and what your thoughts on it are; this will prevent people from telling you things you already know, and help them give their answers at the right level. If this is homework, please add the homework tag; people will still help, so don't worry. – amWhy Nov 9 '12 at 22:25

Hint: In a tree, all edges are bridges. And removing an edge from a cycle of a graph containing a cycle, will not decrease the number of bridges.

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HINT: Removing a bridge increases the number of components by $1$; what is the largest possible number of components of a graph with $k$ vertices? And if you start with just one component, how many increases of $1$ does it take to reach that largest possible number?

Once you’ve got that, you’ll want an example that actually has the largest possible number of bridges; a very simple graph will do the trick.

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Just thinking out loud here: but what if the graph is just defined by a "straight line" of vertices and edges? Then each edge would be a bridge. I think this concept is a bit related to the definition of trees. If this is on the right track, then the answer seems quite obvious. The harder part would be to prove this...

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