Tournaments


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 $ G $ be a tournament with edges colored from a set of three colors. Is it true that $ G $ must have either a rainbow directed cycle of length three or a vertex $ v $ so that every other vertex can be reached from $ v $ by a monochromatic (directed) path?

Keywords: digraph; edge-coloring; tournament

Decomposing an even tournament in directed paths. ★★★

Author(s): Alspach; Mason; Pullman

Conjecture   Every tournament $ D $ on an even number of vertices can be decomposed into $ \sum_{v\in V}\max\{0,d^+(v)-d^-(v)\} $ directed paths.

Keywords:

Edge-disjoint Hamilton cycles in highly strongly connected tournaments. ★★

Author(s): Thomassen

Conjecture   For every $ k\geq 2 $, there is an integer $ f(k) $ so that every strongly $ f(k) $-connected tournament has $ k $ edge-disjoint Hamilton cycles.

Keywords:

Partitionning a tournament into k-strongly connected subtournaments. ★★

Author(s): Thomassen

Problem   Let $ k_1, \dots , k_p $ be positve integer Does there exists an integer $ g(k_1, \dots , k_p) $ such that every $ g(k_1, \dots , k_p) $-strong tournament $ T $ admits a partition $ (V_1\dots , V_p) $ of its vertex set such that the subtournament induced by $ V_i $ is a non-trivial $ k_i $-strong for all $ 1\leq i\leq p $.

Keywords:

Decomposing k-arc-strong tournament into k spanning strong digraphs ★★

Author(s): Bang-Jensen; Yeo

Conjecture   Every k-arc-strong tournament decomposes into k spanning strong digraphs.

Keywords:


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