NOTE The Johnson Graphs Satisfy a Distance Extension Property
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has property \(\) if whenever F and H are connected graphs with \(\) and |H|=|F|+1, and \(\) and \(\) are isometric embeddings, then there is an isometric embedding \(\) such that \(\). It is easy to construct an infinite graph with \(\) for all k, and \(\) holds in almost all finite graphs. Prior to this work, it was not known whether there exist any finite graphs with \(\). We show that the Johnson graphs J(n,3) satisfy \(\) whenever \(\), and that J(6,3) is the smallest graph satisfying \(\). We also construct finite graphs satisfying \(\) and local versions of the extension axioms studied in connection with the Rado universal graph.
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