Tournaments

Graph Theory » Directed Graphs » Tournaments

Monochromatic reachability or rainbow triangles ★★★

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

Posted by mdevos
updated July 15th, 2008

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Graph Theory » Directed Graphs » Tournaments

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:

Posted by fhavet
updated February 26th, 2013

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Graph Theory » Directed Graphs » Tournaments

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:

Posted by fhavet
updated March 8th, 2013

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Graph Theory » Directed Graphs » Tournaments

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 $1\leq i\leq p$ .