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Unordered Love in infinite directed graphs

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  • Peter D. Johnson

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 ν 0 connected both ways to every other vertex, such that D − ν 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.

Suggested Citation

  • Peter D. Johnson, 1992. "Unordered Love in infinite directed graphs," International Journal of Mathematics and Mathematical Sciences, Hindawi, vol. 15, pages 1-4, January.
    • Handle: RePEc:hin:jijmms:594942
    DOI: 10.1155/S0161171292000978
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