Facial Distance Patterns in Planar Graphs
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Statement
For a designated face of an undirected unweighted planar graph, how many distinct distance patterns can vertices have? Li and Parter (STOC 2019) proved an upper bound; Mozes, Wallheimer and Weimann conjectured the true answer matches their lower bound. Proved, closing the gap.
Context
Three immediate consequences follow for undirected unweighted planar graphs: better compression of the Okamura-Seymour metric, less space for constant-time exact distance oracles, and a faster distributed algorithm.
A conjecture with a named home in the planar-graph literature, standing since 2022 against a STOC upper bound and current enough to be posed at a Dagstuhl seminar, with downstream consequences for distance oracles.
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"The simple proof was found by OpenAI's GPT 5.6-Sol model", and the paper says it came from a single prompt describing the state of the art and asking for any improvement on the upper bound. The authors are candid about what that means: "It is surprising (not to say embarrasing) that this open problem has such a simple proof, which has eluded the community despite the human efforts invested in it." Section 2 of the paper is titled The ChatGPT Proof.
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