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Unordered Love in infinite directed graphs

International Journal of Mathematics and Mathematical Sciences
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A digraph D=(V,A) has the Unordered Love Property (ULP) if any two different vertices have a unique common outneighbor. If both (V,A) and (V,A−1) have the ULP, we say that D has the SDULP.A love-master in D is a vertex ν0 connected both ways to every other vertex, such that D−ν0 is a disjoint union of directed cycles.The following results, more or less well-known for finite digraphs, are proven here for D infinite: (i) if D is loopless and has the SDULP, then either D has a love-master, or D is associable with a projective plane, obtainable by taking V as the set of points and the sets of outneighbors of vertices as the lines; (ii) every projective plane arises from a digraph with the SDULP, in this way.

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