Coloring tournaments with few colors: Algorithms and complexity

TL;DR

A new efficient decomposition lemma for tournaments is presented, which is used to design polynomial-time algorithms to color various classes of tournaments with few colors, notably, to color a 2-colorable tournament with ten colors.

Abstract

A $k$-coloring of a tournament is a partition of its vertices into $k$ acyclic sets. Deciding if a tournament is 2-colorable is NP-hard. A natural problem, akin to that of coloring a 3-colorable graph with few colors, is to color a 2-colorable tournament with few colors. This problem does not seem to have been addressed before, although it is a special case of coloring a 2-colorable 3-uniform hypergraph with few colors, which is a well-studied problem with super-constant lower bounds. We present a new efficient decomposition lemma for tournaments, which we use to design polynomial-time algorithms to color various classes of tournaments with few colors, notably, to color a 2-colorable tournament with ten colors. We also use this lemma to prove equivalence between the problems of coloring 3-colorable tournaments and coloring 3-colorable graphs with constantly many colors. For the classes of tournaments considered, we complement our upper bounds with strengthened lower bounds, painting a comprehensive picture of the algorithmic and complexity aspects of coloring tournaments.

Figures (5)

• Figure 1: A path decomposition of $T$. The red arcs $(e_i)$ form a shortest path from $v_0$ to $v_k$, thus all the arcs not depicted between the $v_i$'s go backwards. All the vertices in a given $D_i$ are colored from the color palette indicated by the color of the $D_i$. Notice that because there are no long forward arcs between the $D_i$'s, all arcs between $D_i$'s that share a color palette are backwards.
• Figure 2: Construction of $T$ from a 3-uniform hypergraph ${\cal H}$. (The full description of the construction is in the Proof of Theorem \ref{['thm:2-hardness']}.) The downwards edges in red are drawn only for vertex $v_1$ in ${\cal H}$, but there is an arc from any vertex $v'_{a,i}$ towards all vertices $v_{a,j}$ for any $j$. The remaining arcs all go upwards from the vertices $v_{a,i}$ towards the vertices $v'_{b,j}$ for $a \neq b$.
• Figure 3: Construction of the tournament $U$ from a graph $G$ on five vertices. The dashed red edges are those present in $G$ and all go backwards, whereas the remaining edges are blue and go forwards.
• Figure 4: A 3-chromatic light tournament.
• Figure 5: Construction of $T$ from a 3-uniform hypergraph ${\cal H}$. There is a downwards arc between $v_b'$ and all vertices $v_{b,i}$ for every $b,i$. These are the colored arcs in the figure. All remaining arcs all go upwards from the vertices $v_{a,i}$ towards the vertices $v_{b}'$ for $a \neq b$.

Theorems & Definitions (67)

• Definition 2.1
• Definition 2.2
• Definition 2.3
• Lemma 2.4
• proof
• Lemma 2.5
• proof
• Lemma 2.6
• proof
• Theorem 3.1