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We study the communication complexity of a number of graph properties where the edges of the graph G are distributed between Alice and Bob (i.e., each receives some of the edges as input). Our main results are: * An Omega(n) lower bound on the quantum communication complexity of deciding whether an n-vertex graph G is connected, nearly matching the trivial classical upper bound of O(n log n) bits of communication. * A deterministic upper bound of O(n^3/2log n) bits for deciding if a bipartitearXiv:1204.4596v1 fatcat:3cui5wdo7bekzokvakb3xwvihe