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tournament
PTAS for feedback arc set in tournaments ★★
Question Is there a polynomial time approximation scheme for the feedback arc set problem for the class of tournaments?
Keywords: feedback arc set; PTAS; tournament
Monochromatic reachability or rainbow triangles ★★★
Author(s): Sands; Sauer; Woodrow
In an edge-colored digraph, we say that a subgraph is rainbow if all its edges have distinct colors, and monochromatic if all its edges have the same color.
Problem Let
be a tournament with edges colored from a set of three colors. Is it true that
must have either a rainbow directed cycle of length three or a vertex
so that every other vertex can be reached from
by a monochromatic (directed) path?
![$ G $](/files/tex/b8e7ad0330f925492bf468b5c379baec88cf1b3d.png)
![$ G $](/files/tex/b8e7ad0330f925492bf468b5c379baec88cf1b3d.png)
![$ v $](/files/tex/96cbd9a16c6a5eab03815b093b08f3b2db614e9a.png)
![$ v $](/files/tex/96cbd9a16c6a5eab03815b093b08f3b2db614e9a.png)
Keywords: digraph; edge-coloring; tournament
Monochromatic reachability in edge-colored tournaments ★★★
Author(s): Erdos
Problem For every
, is there a (least) positive integer
so that whenever a tournament has its edges colored with
colors, there exists a set
of at most
vertices so that every vertex has a monochromatic path to some point in
?
![$ n $](/files/tex/ec63d7020a64c039d5f6703b8fa3ab7393358b5b.png)
![$ f(n) $](/files/tex/9579fe06c51fc31a993cd148e8bbc3cb07df464e.png)
![$ n $](/files/tex/ec63d7020a64c039d5f6703b8fa3ab7393358b5b.png)
![$ S $](/files/tex/d2b76a0ee5465d3e3ecc846c8e3d632edd8b2bbf.png)
![$ f(n) $](/files/tex/9579fe06c51fc31a993cd148e8bbc3cb07df464e.png)
![$ S $](/files/tex/d2b76a0ee5465d3e3ecc846c8e3d632edd8b2bbf.png)
Keywords: digraph; edge-coloring; tournament
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