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Cut covers of acyclic digraphs

Maximilian Krone

Abstract

A cut in a digraph $D=(V,A)$ is a set of arcs $\{uv \in A: u\in U, v\notin U\}$, for some $U\subseteq V$. It is known that the arc set $A$ is covered by $k$ cuts if and only if it admits a $k$-coloring such that no two consecutive arcs $uv, vw$ receive the same color. Alon, Bollobás, Gyárfás, Lehel and Scott (2007) observed that every acyclic digraph of maximum indegree at most $\binom{k}{\lfloor k/2 \rfloor}-1$ is covered by $k$ cuts. We prove that this degree condition is best possible (if an enormous outdegree is allowed). Notably, for $k\geq 5$, powers of directed paths do not suffice as extremal examples. Instead, we locate the maximum $d$ such that the $d$-th power of an arbitrarily long directed path is covered by $k$ cuts between $(1-o(1)) \frac{1}{e} 2^k$ and $\frac{1}{2}2^k-2$. Let $k\geq 3$ and $D$ be an acyclic digraph that is not covered by $k$ cuts. We prove that the decision problem whether a digraph that admits a homomorphism to $D$ is covered by $k$ cuts is NP-complete. If $k=3$ and $D$ is the third power of the directed path on 12 vertices, then even the restriction to planar digraphs of maximum indegree and outdegree $3$ holds.

Cut covers of acyclic digraphs

Abstract

A cut in a digraph is a set of arcs , for some . It is known that the arc set is covered by cuts if and only if it admits a -coloring such that no two consecutive arcs receive the same color. Alon, Bollobás, Gyárfás, Lehel and Scott (2007) observed that every acyclic digraph of maximum indegree at most is covered by cuts. We prove that this degree condition is best possible (if an enormous outdegree is allowed). Notably, for , powers of directed paths do not suffice as extremal examples. Instead, we locate the maximum such that the -th power of an arbitrarily long directed path is covered by cuts between and . Let and be an acyclic digraph that is not covered by cuts. We prove that the decision problem whether a digraph that admits a homomorphism to is covered by cuts is NP-complete. If and is the third power of the directed path on 12 vertices, then even the restriction to planar digraphs of maximum indegree and outdegree holds.

Paper Structure

This paper contains 4 sections, 13 equations, 7 figures.

Table of Contents

  1. Cut covers of acyclic digraphs of bounded indegree
  2. Cut covers of acyclic digraphs with vertices of bounded indegree and vertices of bounded outdegree
  3. Complexity of the cut cover problem in acyclic digraphs
  4. Appendix

Figures (7)

  • Figure 1: A 3-coloring of the arc set as in (ii). In this example, each color sets is a cut itself, while in general, it is only contained in some cut. The characteristic sets can be chosen as $\dots\, ,\,\{1,3\}\, ,\,\{1\}\, ,\,\{2,3\}\, ,\,\{2\}\, ,\,\{1,3\}\, ,\,\{1\}\, ,\,\{2,3\}\, ,\,\{2\}\, ,\,\{1,3\}\, ,\,\{1\}\, ,\,\dots$
  • Figure 2: The possible values of $c(\Delta^-,\Delta^+)$ according to Theorem \ref{['or']}.\ref{['Three bounds']}. Each point corresponds to a pair $(\Delta^-,\Delta^+)$. Bound (ii) is stronger than bound (iii) in most cases, except for even $k$ and $\Delta^-\approx \Delta^+$.
  • Figure 3: The values of $c(\Delta^-,\Delta^+)$. Each point corresponds to a pair $(\Delta^-,\Delta^+)$. The values in the grey area have not been determined yet, but are either $5$ or $6$.
  • Figure 4: The digraph $D\in\mathcal{D}(4,3)$ that is not covered by $4$ cuts. The tall arrows indicate that there are all possible arcs from the left set to the right set. We find specific characteristic sets in the subdigraphs $D^-$ and $D^+$.
  • Figure 5: Planar embeddings of the two versions of $P$.
  • ...and 2 more figures