Prove that every undirected finite graph with vertex degree of at least 2 has a cycle.
Intuition-wise i need to prove that there's at least one 'tight -connection'. In other words, Proving that 2 vertexes {$v_i,v_{i+1}$}∈E exists so there's a way to reach from $v_i$ to $v_{i+1}$ and the other way around.
But since i am pretty new to graph theory, I'm not sure i'm even thinking about the question right. Any help would be nice.
Best Answer
Start anywhere, and follow the edges. You'll always be able to continue, since each vertex has degree at least 2, so there will be an unused edge to exit on. If you ever return to a vertex where you've been, you've got a cycle. Otherwise, if the graph is finite, eventually you'll run out of unused vertices and you'll have to revisit some vertex.