# 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 $n$.

## 8 Citations

Nonrepetitive Colourings of Planar Graphs with O(log n) Colours

- MathematicsElectron. J. Comb.
- 2013

A $O(\log n)$ upper bound is proved, which proves that planar graphs have bounded nonrepetitive chromatic number and not just the number of vertex colours.

Notes on Nonrepetitive Graph Colouring

- MathematicsElectron. J. Comb.
- 2008

It is proved that every graph with treewidth $k$ and maximum degree $\Delta$ has a $O(k\Delta)$-colouring that is nonrepetitive on paths, and a “O( k\Delta^3)”-colours that isNonre petitive on walks.

Nonrepetitive graph colouring

- MathematicsThe Electronic Journal of Combinatorics
- 2021

The goal is to give a unified and comprehensive presentation of the major results and proof methods, as well as to highlight numerous open problems about nonrepetitive colourings of graphs.

Nonrepetitive colouring via entropy compression

- MathematicsComb.
- 2016

The proofs of both these results are based on the Moser-Tardos entropy-compression method, and a recent extension by Grytczuk, Kozik and Micek for the nonrepetitive choosability of paths.

Characterisations and examples of graph classes with bounded expansion

- MathematicsEur. J. Comb.
- 2012

Connection Matrices and the Definability of Graph Parameters

- Computer ScienceLog. Methods Comput. Sci.
- 2012

The Finite Rank Theorem for connection matrices of graph parameters definable in Monadic Second Order Logic with counting (CMSOL) is extended and proved in detail and its vast applicability is demonstrated in simplifying known and new non-definability results of graph properties.

Completeness in the Polynomial-Time Hierarchy A Compendium ∗

- Mathematics
- 2008

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…

## References

SHOWING 1-10 OF 14 REFERENCES

ON SQUARE-FREE VERTEX COLORINGS OF GRAPHS

- Mathematics
- 2007

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

- MathematicsRandom Struct. Algorithms
- 2002

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

- MathematicsArs Comb.
- 2004

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

- MathematicsInt. J. Math. Math. Sci.
- 2007

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

- MathematicsSIAM J. Comput.
- 1981

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 ∗

- Mathematics
- 2008

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

- Computer Science
- 2001

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.

Estimation of sparse hessian matrices and graph coloring problems

- Computer ScienceMath. Program.
- 1984

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.