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Omni-MATH / A tournament is a directed graph for which every (unordered) pair of vertices…

Problem

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problem

A tournament is a directed graph for which every (unordered) pair of vertices has a single directed edge from one vertex to the other. Let us define a proper directed-edge-coloring to be an assignment of a color to every (directed) edge, so that for every pair of directed edges

\overrightarrow{uv}

and

\overrightarrow{vw}

, those two edges are in different colors. Note that it is permissible for

\overrightarrow{uv}

and

\overrightarrow{uw}

to be the same color. The directed-edge-chromatic-number of a tournament is defined to be the minimum total number of colors that can be used in order to create a proper directed-edge-coloring. For each

n

, determine the minimum directed-edge-chromatic-number over all tournaments on

n

vertices.

Plain-text mathematical notation (without MathML)

A tournament is a directed graph for which every (unordered) pair of vertices has a single directed edge from one vertex to the other.  Let us define a proper directed-edge-coloring to be an assignment of a color to every (directed) edge, so that for every pair of directed edges (uv)→ and (vw)→, those two edges are in different colors.  Note that it is permissible for (uv)→ and (uw)→ to be the same color.  The directed-edge-chromatic-number of a tournament is defined to be the minimum total number of colors that can be used in order to create a proper directed-edge-coloring.  For each n, determine the minimum directed-edge-chromatic-number over all tournaments on n vertices.

Original LaTeX notation

A tournament is a directed graph for which every (unordered) pair of vertices has a single directed edge from one vertex to the other.  Let us define a proper directed-edge-coloring to be an assignment of a color to every (directed) edge, so that for every pair of directed edges $\overrightarrow{uv}$ and $\overrightarrow{vw}$, those two edges are in different colors.  Note that it is permissible for $\overrightarrow{uv}$ and $\overrightarrow{uw}$ to be the same color.  The directed-edge-chromatic-number of a tournament is defined to be the minimum total number of colors that can be used in order to create a proper directed-edge-coloring.  For each $n$, determine the minimum directed-edge-chromatic-number over all tournaments on $n$ vertices.

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task_id: e4066512-8101-598d-979a-1be4772b7d78
task_key: test--e4066512-8101-598d-979a-1be4772b7d78
task_revision_id: 4
config / split: / test

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