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Sands, Bill
Monochromatic reachability in arc-colored digraphs ★★★
Author(s): Sands; Sauer; Woodrow
Conjecture For every 
, there exists an integer 
such that if 
is a digraph whose arcs are colored with colors, then has a 
set which is the union of stables sets so that every vertex has a monochromatic path to some vertex in .
, there exists an integer such that if is a digraph whose arcs are colored with colors, then has a set which is the union of stables sets so that every vertex has a monochromatic path to some vertex in . Keywords:
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?
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? Keywords: digraph; edge-coloring; tournament
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