Geodesic
Touching perfect matchings and halving lines
Ars Mathematica Contemporanea, Tome 15 (2018) no. 2, pp. 375-382.

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Let V be a set of 2m (1 ≤ m  ∞) points in the plane. Two segments I, J with endpoints in V cross if relint I ∩ relint J is a singleton. A (perfect) cross-matching M on V is a set of m segments with endpoints in V such that every two segments in M cross. A halving line of V is a line l spanned by two points of V such that each one of the two open half planes bounded by l contains fewer than m points of V. Pach and Solymosi proved that if V is in general position, then V admits a perfect cross-matching iff V has exactly m halving lines. The aim of this note is to extend this result to the general case (where V is unrestricted).
DOI : 10.26493/1855-3974.1228.d7d
Keywords: Bigraphs, cross-matching, halving lines, perfect matchings 
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Micha A. Perles; Horst Martini; Yaakov S. Kupitz. Touching perfect matchings and halving lines. Ars Mathematica Contemporanea, Tome 15 (2018) no. 2, pp. 375-382. doi : 10.26493/1855-3974.1228.d7d. http://geodesic.mathdoc.fr/articles/10.26493/1855-3974.1228.d7d/

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