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Antiflows, oriented and strong oriented colorings of graphs

Robert Šámal — 2004

Archivum Mathematicum

We present an overview of the theory of nowhere zero flows, in particular the duality of flows and colorings, and the extension to antiflows and strong oriented colorings. As the main result, we find the asymptotic relation between oriented and strong oriented chromatic number.

On the number of intersections of two polygons

Jakub ČernýJan KáraDaniel KráľPavel PodbrdskýMiroslava SotákováRobert Šámal — 2003

Commentationes Mathematicae Universitatis Carolinae

We study the maximum possible number f ( k , l ) of intersections of the boundaries of a simple k -gon with a simple l -gon in the plane for k , l 3 . To determine the number f ( k , l ) is quite easy and known when k or l is even but still remains open for k and l both odd. We improve (for k l ) the easy upper bound k l - l to k l - k / 6 - l and obtain exact bounds for k = 5 ( f ( 5 , l ) = 4 l - 2 ) in this case.

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