On the facial Thue choice index of plane graphs

Schreyer, Jens; Škrabuľáková, Erika

doi:10.1016/j.disc.2012.01.023

Cited by 15 publications

(21 citation statements)

References 9 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The best known upper bound is O(log n) where n is the number of vertices, due to Dujmović, Frati, Joret, and Wood [16]. Note that several works have studied colourings of planar graphs in which only facial paths are required to be nonrepetitively coloured [4,8,33,34,44,45,48].…”

Section: Introductionmentioning

confidence: 99%

Planar graphs have bounded nonrepetitive chromatic number

Dujmović

Esperet

Joret

et al. 2020

AiC

View full text Add to dashboard Cite

A colouring of a graph is nonrepetitive if for every path of even order, the sequence of colours on the first half of the path is different from the sequence of colours on the second half. We show that planar graphs have nonrepetitive colourings with a bounded number of colours, thus proving a conjecture of Alon, Grytczuk, Hałuszczak and Riordan (2002). We also generalise this result for graphs of bounded Euler genus, graphs excluding a fixed minor, and graphs excluding a fixed topological minor.

show abstract

Section: Introductionmentioning

confidence: 99%

Planar graphs have bounded nonrepetitive chromatic number

Dujmović

Esperet

Joret

et al. 2020

AiC

View full text Add to dashboard Cite

show abstract

“…The least integer k such that for every list assignment

L : E \to 2^{double-struckN}

with

| L (e) | \geq k

there exists a facial nonrepetitive edge coloring

c : E \to N

where

c (e) \in L (e)

for every

e \in E

is called the facial Thue choice index of G and is denoted by

π_{f l}^{'} (G)

. For this parameter, a much worse upper bound asserting that

π_{f l}^{'} (G) \leq 291

for every plane graph was settled by means of the Lovász Local Lemma by Schreyer and Škrabul'áková . In this article, we improve this upper bound using entropy‐compression method, and prove the following.…”

Section: Introductionmentioning

confidence: 99%

On the Facial Thue Choice Index via Entropy Compression

Przybyło

2013

Journal of Graph Theory

View full text Add to dashboard Cite

A sequence is nonrepetitive if it contains no identical consecutive subsequences. An edge coloring of a path is nonrepetitive if the sequence of colors of its consecutive edges is nonrepetitive. By the celebrated construction of Thue, it is possible to generate nonrepetitive edge colorings for arbitrarily long paths using only three colors. A recent generalization of this concept implies that we may obtain such colorings even if we are forced to choose edge colors from any sequence of lists of size 4 (while sufficiency of lists of size 3 remains an open problem). As an extension of these basic ideas, Havet, Jendrol', Soták, and Škrabul'áková proved that for each plane graph, eight colors are sufficient to provide an edge coloring so that every facial path is nonrepetitively colored. In this article, we prove that the same is possible from lists, provided that these have size at least 12. We thus improve the previous bound of 291 (proved by means of the Lovász Local Lemma). Our approach is based on the Moser–Tardos entropy‐compression method and its recent extensions by Grytczuk, Kozik, and Micek, and by Dujmović, Joret, Kozik, and Wood.

show abstract

“…The problem is also well-studied in the nonrepetitive setting [18,24,25,29,35,46,47,48]. A nonrepetitive vertex L-colouring of G is a nonrepetitive vertex colouring c of G in which c(v) ∈ L(v) for every vertex v ∈ V (G).…”

Section: List Colouringsmentioning

confidence: 99%

“…Using the Moser and Tardos technique, Schreyer andŠkrabul'áková [47] proved that a constant bound also exists for the facial Thue choice index -the smallest l such that in any l-list assignment of the edges of a graph G, a facial nonrepetitive edge colouring of G exists, which is denoted π f,ch (G). They found an upper bound of 291 for this parameter, which was later improved to 12 by Przyby lo [46], and most recently to 10 by Gonçalves et al [25].…”

Section: Facial Edge Colouringsmentioning

confidence: 99%

See 1 more Smart Citation

Facial Nonrepetitive Graph Colourings

Rioux-Maldague¹

View full text Add to dashboard Cite

Given a plane graph G, a facial path is a path of consecutive vertices on the boundary walk of a face of G. A nonrepetitive facial colouring of G is a vertex colouring such that the sequence of colours of any facial path of G is nonrepetitive, and the minimum number of colours required for such a colouring is the facial Thue chromatic number of G.Using a new blocking set technique, we show that the facial Thue chromatic number of an outerplane graph is bounded by 11, and by 7 for outerplane graphs that contain at most one 2-connected component. Furthermore, we show that the facial Thue chromatic number of plane graphs is bounded by twice the facial Thue chromatic number of outerplane graphs, which results in an upper bound of 22 for this parameter, an improvement over the previous bound of 24 by Barat and Czap

show abstract

On the facial Thue choice index of plane graphs

Cited by 15 publications

References 9 publications

Planar graphs have bounded nonrepetitive chromatic number

Planar graphs have bounded nonrepetitive chromatic number

On the Facial Thue Choice Index via Entropy Compression

Facial Nonrepetitive Graph Colourings

Contact Info

Product

Resources

About