(Solved): 1. Answer the following questions:(You should prove it or disprove it. Disproof could be a countere ...

1. Answer the following questions:(You should prove it or disprove it. Disproof could be a counterexample.) (30 marks) (a) A graph \( G \) has at least \( C_{\chi(G)}^{2} \) edges. (6 marks) (b) Let \( G \) be a connected n-vertex graph, \( G \) has exactly one cycle if and only if \( G \) has exactly \( n \) edges. (6 marks) (c) A group \( G \) cannot be a union of two proper subgroups. \( (6 \) marks)