Total Thue colourings of graphs

Schreyer, Jens; Škrabuľáková, Erika

doi:10.1007/s40879-014-0016-2

Cited by 9 publications

(18 citation statements)

References 18 publications

(26 reference statements)

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“…The notion of Total Thue coloring was introduced by Schreyer andŠkrabuvláková [17]. A coloring of the edges and the vertices of a graph is a weak total Thue coloring if the sequence of consecutive vertex-colors and edge-colors of every path is non repetitive.…”

Section: Introductionmentioning

confidence: 99%

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Another Approach to Non-Repetitive Colorings of Graphs of Bounded Degree

Rosenfeld

2020

Electron. J. Combin.

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We propose a new proof technique that applies to the same problems as the Lovász Local Lemma or the entropy-compression method. We present this approach in the context of non-repetitive colorings and we use it to improve upper-bounds relating different non-repetitive chromatic numbers to the maximal degree of a graph. It seems that there should be other interesting applications of the presented approach. In terms of upper-bounds our approach seems to be as strong as entropy-compression, but the proofs are more elementary and shorter. The applications we provide in this paper are upper bounds for graphs of maximal degree at most $\Delta$: a minor improvement on the upper-bound of the non-repetitive chromatic number, a $4.25\Delta +o(\Delta)$ upper-bound on the weak total non-repetitive chromatic number, and a $ \Delta^2+\frac{3}{2^{1/3}}\Delta^{5/3}+ o(\Delta^{5/3})$ upper-bound on the total non-repetitive chromatic number of graphs. This last result implies the same upper-bound for the non-repetitive chromatic index of graphs, which improves the best known bound.

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Section: Introductionmentioning

confidence: 99%

“…These two parameters both have there list-coloring counterpart denoted respectively by π Twch (G) and π Tch (G). Schreyer andŠkrabuvláková [17] showed that π T (G) 5∆ 2 +o(∆ 2 ), π Tch (G) 17.9856∆ 2 and π Tw (G) |E(G)| − |V (G)| + 5. We remark that the second bound also relies on an application of the Lovász Local Lemma.…”

Section: Introductionmentioning

confidence: 99%

Another Approach to Non-Repetitive Colorings of Graphs of Bounded Degree

Rosenfeld

2020

Electron. J. Combin.

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“…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%

“…A total colouring is a colouring in which both edges and vertices of a graph are coloured such that no adjacent edges, and no edge and its endpoints, are assigned the same colour. Such colourings were also studied in the nonrepetitive setting by Schreyer andŠkrabul'áková [48]. They defined two versions of the problem.…”

Section: Total Thue Chromatic Numbermentioning

confidence: 99%

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Facial Nonrepetitive Graph Colourings

Rioux-Maldague¹

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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

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