Is a circuit-free graph with n vertices and at least edges connected? Why?
Whether a circuit free graph with vertices and at least edges is connected or not.
A circuit free graph with vertices and at least edges.
Any tree with vertices has
Suppose is a circuit free graph with vertices and edges. Let be the connected component of . To show that is connected, we will show that .
Each graph is a tree since each is connected and circuit free.
For each have vertices. Then,
For a positive integer any tree with vertices has edges.
Thus, has edges.
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