• Corpus ID: 5489770

The complexity of nonrepetitive edge coloring of graphs

@article{Manin2007TheCO,
  title={The complexity of nonrepetitive edge coloring of graphs},
  author={Fedor Manin},
  journal={ArXiv},
  year={2007},
  volume={abs/0709.4497}
}
A squarefree word is a sequence $w$ of symbols such that there are no strings $x, y$, and $z$ for which $w=xyyz$. A nonrepetitive coloring of a graph is an edge coloring in which the sequence of colors along any open path is squarefree. We show that determining whether a graph $G$ has a nonrepetitive $k$-coloring is $\Sigma_2^p$-complete. When we restrict to paths of lengths at most $n$, the problem becomes NP-complete for fixed $n$. 

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References

SHOWING 1-10 OF 14 REFERENCES
The complexity of nonrepetitive coloring
ON SQUARE-FREE VERTEX COLORINGS OF GRAPHS
A sequence of symbols a 1 , a 2 … is called square-free if it does not contain a subsequence of consecutive terms of the form x 1 , …, x m , x 1 , …, x m . A century ago Thue showed that there exist
Nonrepetitive colorings of graphs
TLDR
This paper considers a natural generalization of Thue's sequences for colorings of graphs and shows that the Thue number stays bounded for graphs with bounded maximum degree, in particular, π (G) ≤ cΔ(G)2 for some absolute constant c.
Square-free colorings of graphs
TLDR
The (walk) Thue number of complete multipartite graphs is bounded which in turn gives a bound for arbitrary graphs in general and for perfect graphs in particular.
Nonrepetitive Colorings of Graphs - A Survey
A vertex coloring f of a graph G is nonrepetitive if there are no integer r≥1 and a simple path v1,…,v2r in G such that f(vi)=f(vr
The NP-Completeness of Edge-Coloring
TLDR
It is shown that it is NP-complete to determine the chromatic index of an arbitrary graph, even for cubic graphs.
Completeness in the Polynomial-Time Hierarchy A Compendium ∗
We present a Garey/Johnson-style list of problems known to be complete for the second and higher levels of the polynomial-time Hierarchy (polynomial hierarchy, or PH for short). We also include the
Completeness in the Polynomial Time Hierarchy
TLDR
This report is trying to compile a list of problems that reside in the polynomial hieararchy above the second level, and does not contain some recent results, nor does it list any of the results on petri-nets, non-monotonic logics, and databases.
On square-free edge colorings of graphs
Estimation of sparse hessian matrices and graph coloring problems
TLDR
This work approaches the problem of estimating Hessian matrices by differences from a graph theoretic point of view and shows that both direct and indirect approaches have a natural graph coloring interpretation.
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