International Journal of Mathematics and Mathematical Sciences
Volume 15 (1992), Issue 4, Pages 753-756
doi:10.1155/S0161171292000978
Unordered Love in infinite directed graphs
Department of Algebra, Combinatorics, and Analysis, Auburn University, Auburn 36849, Alabama, USA
Received 4 September 1990
Copyright © 1992 Peter D. Johnson. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Abstract
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.