Determining the communication complexity of graph connectivity has been one of the longest standing open problems in communication complexity since the seminal work of Babai, Frankl, and Simon [FOCS'86]. A simple protocol transmits a spanning forest using $O(N\log N)$ communication on $N$-vertex graphs. While this was shown optimal for deterministic protocols by Hajnal, Maass, and Turán [STOC'88], the optimal nondeterministic and randomized complexities have remained unresolved for decades. We settle the nondeterministic communication complexity of graph connectivity by proving an optimal $Ω(N\log N)$ lower bound. The best prior bound was $Ω(N\log\log N)$ due to Raz and Spieker [Combinatorica'95]. Our bound is shown by upper bounding the maximum size of $1$-monochromatic rectangles. Generalizing our techniques to weighted, almost $1$-monochromatic rectangles yields an $Ω(N \cdot \log(\min(1/\varepsilon,N)))$ lower bound for randomized protocols with error probability $\varepsilon$. In particular, this bound is optimal for any polynomially small $\varepsilon$. For constant error probability, we also prove that any $r$-round protocol requires $Ω(N/r \cdot \log N)$ communication, which is optimal for every constant $r$. Previously, optimal $Ω(N\log N)$ lower bounds for graph connectivity were known only for $1$-round [Sun, Woodruff; APPROX/RANDOM'15] and $2$-round protocols [Radhakrishnan, Reddy, Venkat; ITCS'26]. Our proof introduces a new communication problem which can be viewed as a composition of inner product and pointer chasing problems. Our lower bounds can be extended to other fundamental graph properties, and also imply optimality of known query and streaming algorithms for related problems.

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