Unordered Love in infinite directed graphs
Peter D. Johnson · International Journal of Mathematics and Mathematical Sciences · 1992
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.