(1) Let G = (V, E) be an undirected graph with d components, where |V| = b and |E| =c  . Please prove that c ≥ b − d . (10%)

(2) Consider the complete bipartite graph ?&,! , where the vertex set is partitioned into two disjoint sets ?# and ?$ with |?# | = ? and |?$ | = ?. Derive the necessary and sufficient conditions for ? and ? such that ?&,! contains an Eulerian path but NO Eulerian circuit. (Consider all possible cases for ? and ? ). (10%)