- 期刊
UNORDERED LOVE IN INFINITE DIRECTED GRAPHS
摘要
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 0 connected both ways to every other vertex, such that D- v_0 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.