Nonrepetitive Graph Colouring

Wood, David R.

doi:10.37236/9777

Cited by 11 publications

(7 citation statements)

References 128 publications

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“…Let us mention that the notion of nonrepetitive coloring of graphs, as introduced by Alon, Hałuszczak, Grytczuk, and Riordan in [1], can be considered more generally, in a way similar to the usual proper coloring of graphs (in both, edge or vertex version). A recent survey by Wood [20] collects many interesting results on this topic.…”

Section: Theorem 4 (Whitney 1932) a Graph G With At Least Two Edges I...mentioning

confidence: 99%

Strongly proper connected coloring of graphs

Dębski¹,

Grytczuk²,

Naroski³

et al. 2023

Preprint

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We study a new variant of connected coloring of graphs based on the concept of strong edge coloring (every color class forms an induced matching). In particular, an edge-colored path is strongly proper if its color sequence does not contain identical terms within a distance of at most two. A strong proper connected coloring of G is the one in which every pair of vertices is joined by at least one strongly proper path. Let spc(G) denote the least number of colors needed for such coloring of a graph G.We prove that the upper bound spc(G) ≤ 5 holds for any 2-connected graph G. On the other hand, we demonstrate that there are 2-connected graphs with arbitrarily large girth satisfying spc(G) ≥ 4. Additionally, we prove that graphs whose cycle lengths are divisible by 3 satisfy spc(G) ≤ 3.We also consider briefly other connected colorings defined by various restrictions on color sequences of connecting paths. For instance, in a nonrepetitive connected coloring of G, every pair of vertices should be joined by a path whose color sequence is nonrepetitive, that is, it does not contain two adjacent identical blocks. We demonstrate that 4-connected graphs are 6-colorable in the nonrepetitive sense. A similar conclusion with a finite upper bound on the number of colors holds for a much wider variety of connected colorings. We end the paper with a general conjecture stating that a similar statement holds for 2-connected graphs.

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Section: Theorem 4 (Whitney 1932) a Graph G With At Least Two Edges I...mentioning

confidence: 99%

Strongly proper connected coloring of graphs

Dębski¹,

Grytczuk²,

Naroski³

et al. 2023

Preprint

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show abstract

“…The nonrepetitive chromatic number, π(G), is the minimum number of colours in a nonrepetitive colouring of G. Nonrepetitive colourings were introduced by Alon, Grytczuk, Hałuszczak, and Riordan [1] and have since been widely studied; see the survey [29].…”

Section: We Now Give An Application Of Theoremmentioning

confidence: 99%

The product structure of squaregraphs

Hickingbotham¹,

Jungeblut²,

Merker³

et al. 2022

Preprint

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A squaregraph is a plane graph in which each internal face is a 4-cycle and each internal vertex has degree at least 4. This paper proves that every squaregraph is isomorphic to a subgraph of the semi-strong product of an outerplanar graph and a path. We generalise this result for infinite squaregraphs, and show that this is best possible in the sense that "outerplanar graph" cannot be replaced by "forest".

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“…Nonrepetitive colourings were introduced by Alon, Grytczuk, Hałuszczak, and Riordan [2] and have since been well-studied (see the survey [77]…”

Section: Nonrepetitive Colouringsmentioning

confidence: 99%

Shallow Minors, Graph Products and Beyond Planar Graphs

Hickingbotham¹,

Wood²

2021

Preprint

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The planar graph product structure theorem of Dujmović, Joret, Micek, Morin, Ueckerdt, and Wood [J. ACM 2020] states that every planar graph is a subgraph of the strong product of a graph with bounded treewidth and a path. This result has been the key tool to resolve important open problems regarding queue layouts, nonrepetitive colourings, centered colourings, and adjacency labelling schemes. In this paper, we extend this line of research by utilizing shallow minors to prove analogous product structure theorems for several beyond planar graph classes. The key observation that drives our work is that many beyond planar graphs can be described as a shallow minor of the strong product of a planar graph with a small complete graph. In particular, we show that power of planar graphs, k-planar, (k, p)-cluster planar, k-semi-fan-planar graphs and k-fan-bundle planar graphs can be described in this manner. Using a combination of old and new results, we deduce that these classes have bounded queuenumber, bounded nonrepetitive chromatic number, polynomial p-centred chromatic numbers, linear strong colouring numbers, and cubic weak colouring numbers. In addition, we show that k-gap planar graphs have super-linear local treewidth and, as a consequence, cannot be described as a subgraph of the strong product of a graph with bounded treewidth and a path.

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Nonrepetitive Graph Colouring

Cited by 11 publications

References 128 publications

Strongly proper connected coloring of graphs

Strongly proper connected coloring of graphs

The product structure of squaregraphs

Shallow Minors, Graph Products and Beyond Planar Graphs

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