New bounds on the classical and quantum communication complexity of some graph properties

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: 1. 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. 2. A deterministic upper bound of O(n^{3/2} log n) bits for deciding if a bipartite graph contains a perfect matching, and a quantum lower bound of Omega(n) for this problem. 3. A Theta(n^2) bound for the randomized communication complexity of deciding if a graph has an Eulerian tour, and a Theta(n^{3/2}) bound for its quantum communication complexity. 4. The first two quantum lower bounds are obtained by exhibiting a reduction from the n-bit Inner Product problem to these graph problems, which solves an open question of Babai, Frankl and Simon [Babai et al 1986]. The third quantum lower bound comes from recent results about the quantum communication complexity of composed functions. We also obtain essentially tight bounds for the quantum communication complexity of a few other problems, such as deciding if $G$ is triangle-free, or if G is bipartite, as well as computing the determinant of a distributed matrix.