scholarly journals Notes on Nonrepetitive Graph Colouring

10.37236/823 ◽  
2008 ◽  
Vol 15 (1) ◽  
Author(s):  
János Barát ◽  
David R. Wood
Keyword(s):  
Maximum Degree ◽  
Maximum Density ◽  
Graph Colouring ◽  
Vertex Colouring

A vertex colouring of a graph is nonrepetitive on paths if there is no path $v_1,v_2,\dots,v_{2t}$ such that $v_i$ and $v_{t+i}$ receive the same colour for all $i=1,2,\dots,t$. We determine the maximum density of a graph that admits a $k$-colouring that is nonrepetitive on paths. We prove that every graph has a subdivision that admits a $4$-colouring that is nonrepetitive on paths. The best previous bound was $5$. We also study colourings that are nonrepetitive on walks, and provide a conjecture that would imply that every graph with maximum degree $\Delta$ has a $f(\Delta)$-colouring that is nonrepetitive on walks. We prove that every graph with treewidth $k$ and maximum degree $\Delta$ has a $O(k\Delta)$-colouring that is nonrepetitive on paths, and a $O(k\Delta^3)$-colouring that is nonrepetitive on walks.A corrigendum was added to this paper on Dec 12, 2014.

scholarly journals Clustered 3-colouring graphs of bounded degree

2021 ◽  
pp. 1-13
Author(s):  
Vida Dujmović ◽  
Louis Esperet ◽  
Pat Morin ◽  
Bartosz Walczak ◽  
David R. Wood
Keyword(s):  
Planar Graphs ◽  
Maximum Degree ◽  
Bounded Degree ◽  
Bounded Number ◽  
Proper Vertex ◽  
Vertex Colouring

Abstract A (not necessarily proper) vertex colouring of a graph has clustering c if every monochromatic component has at most c vertices. We prove that planar graphs with maximum degree $\Delta$ are 3-colourable with clustering $O(\Delta^2)$ . The previous best bound was $O(\Delta^{37})$ . This result for planar graphs generalises to graphs that can be drawn on a surface of bounded Euler genus with a bounded number of crossings per edge. We then prove that graphs with maximum degree $\Delta$ that exclude a fixed minor are 3-colourable with clustering $O(\Delta^5)$ . The best previous bound for this result was exponential in $\Delta$ .

scholarly journals Nonrepetitive Graph Colouring

10.37236/9777 ◽  
2021 ◽  
Vol 1000 ◽  
Author(s):  
David R Wood
Keyword(s):  
Graph Colouring ◽  
Open Problems ◽  
Famous Theorem ◽  
Proof Methods ◽  
Vertex Colouring

A vertex colouring of a graph $G$ is nonrepetitive if $G$ contains no path for which the first half of the path is assigned the same sequence of colours as the second half. Thue's famous theorem says that every path is nonrepetitively 3-colourable. This paper surveys results about nonrepetitive colourings of graphs. 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.

scholarly journals The Distance-t Chromatic Index of Graphs

2013 ◽  
Vol 23 (1) ◽  
pp. 90-101 ◽  
Author(s):  
TOMÁŠ KAISER ◽  
ROSS J. KANG
Keyword(s):  
Maximum Degree ◽  
Upper Bounds ◽  
The Other ◽  
Graph Colouring ◽  
Chromatic Index ◽  
Multiplicative Factor ◽  
Strong Chromatic Index ◽  
Absolute Positive Constant ◽  
Large Maximum ◽  
Fixed Positive Integer

We consider two graph colouring problems in which edges at distance at most t are given distinct colours, for some fixed positive integer t. We obtain two upper bounds for the distance-t chromatic index, the least number of colours necessary for such a colouring. One is a bound of (2-ε)Δt for graphs of maximum degree at most Δ, where ε is some absolute positive constant independent of t. The other is a bound of O(Δt/log Δ) (as Δ → ∞) for graphs of maximum degree at most Δ and girth at least 2t+1. The first bound is an analogue of Molloy and Reed's bound on the strong chromatic index. The second bound is tight up to a constant multiplicative factor, as certified by a class of graphs of girth at least g, for every fixed g ≥ 3, of arbitrarily large maximum degree Δ, with distance-t chromatic index at least Ω(Δt/log Δ).

Graph Colouring with No Large Monochromatic Components

2008 ◽  
Vol 17 (4) ◽  
pp. 577-589 ◽  
Author(s):  
NATHAN LINIAL ◽  
JIŘÍ MATOUŠEK ◽  
OR SHEFFET ◽  
GÁBOR TARDOS
Keyword(s):  
Planar Graphs ◽  
Maximum Degree ◽  
The Other ◽  
Graph Colouring ◽  
Connected Subgraph ◽  
Regular Graphs ◽  
Maximum Order ◽  
Asymptotically Optimal ◽  
Vertex Graph ◽  
Order Of Magnitude

For a graph G and an integer t we let mcct(G) be the smallest m such that there exists a colouring of the vertices of G by t colours with no monochromatic connected subgraph having more than m vertices. Let be any non-trivial minor-closed family of graphs. We show that mcc2(G) = O(n2/3) for any n-vertex graph G ∈ . This bound is asymptotically optimal and it is attained for planar graphs. More generally, for every such , and every fixed t we show that mcct(G)=O(n2/(t+1)). On the other hand, we have examples of graphs G with no Kt+3 minor and with mcct(G)=Ω(n2/(2t−1)).It is also interesting to consider graphs of bounded degrees. Haxell, Szabó and Tardos proved mcc2(G) ≤ 20000 for every graph G of maximum degree 5. We show that there are n-vertex 7-regular graphs G with mcc2(G)=Ω(n), and more sharply, for every ϵ > 0 there exists cϵ > 0 and n-vertex graphs of maximum degree 7, average degree at most 6 + ϵ for all subgraphs, and with mcc2(G) ≥ cϵn. For 6-regular graphs it is known only that the maximum order of magnitude of mcc2 is between $\sqrt n$ and n.We also offer a Ramsey-theoretic perspective of the quantity mcct(G).

scholarly journals NP-completeness and One Polynomial Subclass of the Two-Step Graph Colouring Problem

2019 ◽  
Vol 26 (3) ◽  
pp. 405-419
Author(s):  
Natalya Sergeevna Medvedeva ◽  
Alexander Valeryevich Smirnov
Keyword(s):  
Connected Graph ◽  
Special Interest ◽  
Recognition Problem ◽  
Graph Colouring ◽  
Rectangular Grid ◽  
Grid Graph ◽  
Grid Graphs ◽  
Np Completeness ◽  
Polynomial Reduction ◽  
Vertex Colouring

In this paper, we study the two-step colouring problem for an undirected connected graph. It is required to colour the graph in a given number of colours in a way, when no pair of vertices has the same colour, if these vertices are at a distance of 1 or 2 between each other. Also the corresponding recognition problem is set. The problem is closely related to the classical graph colouring problem. In the article, we study and prove the polynomial reduction of the problems to each other. So it allows us to prove NP-completeness of the problem of two-step colouring. Also we specify some of its properties. Special interest is paid to the problem of two-step colouring in application to rectangular grid graphs. The maximum vertex degree in such a graph is between 0 and 4. For each case, we elaborate and prove the function of two-vertex colouring in the minimum possible number of colours. The functions allow each vertex to be coloured independently from others. If vertices are examined in a sequence, colouring time is polynomial for a rectangular grid graph.

scholarly journals Defective and Clustered Graph Colouring

10.37236/7406 ◽  
2018 ◽  
Vol 1000 ◽  
Author(s):  
David R. Wood
Keyword(s):  
Planar Graphs ◽  
Maximum Degree ◽  
Graph Colouring ◽  
Outerplanar Graphs ◽  
Open Problems ◽  
Arbitrary Graph ◽  
Maximum Average Degree ◽  
Graph Classes ◽  
Small Defect ◽  
A Minor

Consider the following two ways to colour the vertices of a graph where the requirement that adjacent vertices get distinct colours is relaxed. A colouring has defect $d$ if each monochromatic component has maximum degree at most $d$. A colouring has clustering $c$ if each monochromatic component has at most $c$ vertices. This paper surveys research on these types of colourings, where the first priority is to minimise the number of colours, with small defect or small clustering as a secondary goal. List colouring variants are also considered. The following graph classes are studied: outerplanar graphs, planar graphs, graphs embeddable in surfaces, graphs with given maximum degree, graphs with given maximum average degree, graphs excluding a given subgraph, graphs with linear crossing number, linklessly or knotlessly embeddable graphs, graphs with given Colin de Verdière parameter, graphs with given circumference, graphs excluding a given immersion, graphs with given thickness, graphs with given stack- or queue-number, graphs excluding $K_t$ as a minor, graphs excluding $K_{s,t}$ as a minor, and graphs excluding an arbitrary graph $H$ as a minor. Several open problems are discussed.

scholarly journals Defective and clustered choosability of sparse graphs

2019 ◽  
Vol 28 (5) ◽  
pp. 791-810 ◽  
Author(s):  
Kevin Hendrey ◽  
David R. Wood
Keyword(s):  
Graph Theory ◽  
Maximum Degree ◽  
Discrete Math ◽  
Average Degree ◽  
Graph Colouring ◽  
Sparse Graphs ◽  
Trade Off ◽  
Maximum Average Degree ◽  
The Earth

AbstractAn (improper) graph colouring hasdefect dif each monochromatic subgraph has maximum degree at mostd, and hasclustering cif each monochromatic component has at mostcvertices. This paper studies defective and clustered list-colourings for graphs with given maximum average degree. We prove that every graph with maximum average degree less than (2d+2)/(d+2)kisk-choosable with defectd. This improves upon a similar result by Havet and Sereni (J. Graph Theory, 2006). For clustered choosability of graphs with maximum average degreem, no (1-ɛ)mbound on the number of colours was previously known. The above result withd=1 solves this problem. It implies that every graph with maximum average degreemis$\lfloor{\frac{3}{4}m+1}\rfloor$-choosable with clustering 2. This extends a result of Kopreski and Yu (Discrete Math., 2017) to the setting of choosability. We then prove two results about clustered choosability that explore the trade-off between the number of colours and the clustering. In particular, we prove that every graph with maximum average degreemis$\lfloor{\frac{7}{10}m+1}\rfloor$-choosable with clustering 9, and is$\lfloor{\frac{2}{3}m+1}\rfloor$-choosable with clusteringO(m). As an example, the later result implies that every biplanar graph is 8-choosable with bounded clustering. This is the best known result for the clustered version of the earth–moon problem. The results extend to the setting where we only consider the maximum average degree of subgraphs with at least some number of vertices. Several applications are presented.

scholarly journals A Short Proof of the Hajnal–Szemerédi Theorem on Equitable Colouring

2008 ◽  
Vol 17 (2) ◽  
pp. 265-270 ◽  
Author(s):  
H. A. KIERSTEAD ◽  
A. V. KOSTOCHKA
Keyword(s):  
Positive Integer ◽  
Polynomial Time ◽  
Maximum Degree ◽  
Short Proof ◽  
Polynomial Time Algorithm ◽  
Time Algorithm ◽  
Proper Vertex ◽  
Colour Classes ◽  
Vertex Colouring

A proper vertex colouring of a graph is equitable if the sizes of colour classes differ by at most one. We present a new shorter proof of the celebrated Hajnal–Szemerédi theorem: for every positive integer r, every graph with maximum degree at most r has an equitable colouring with r+1 colours. The proof yields a polynomial time algorithm for such colourings.

scholarly journals Bounds for Distinguishing Invariants of Infinite Graphs

10.37236/6362 ◽  
2017 ◽  
Vol 24 (3) ◽  
Author(s):  
Wilfried Imrich ◽  
Rafał Kalinowski ◽  
Monika Pilśniak ◽  
Mohammad Hadi Shekarriz
Keyword(s):  
Chromatic Number ◽  
Lower Bounds ◽  
Maximum Degree ◽  
Infinite Graphs ◽  
Chromatic Index ◽  
Distinguishing Number ◽  
Minimum Number ◽  
Edge Colourings ◽  
Delta G ◽  
Vertex Colouring

We consider infinite graphs. The distinguishing number $D(G)$ of a graph $G$ is the minimum number of colours in a vertex colouring of $G$ that is preserved only by the trivial automorphism. An analogous invariant for edge colourings is called the distinguishing index, denoted by $D'(G)$. We prove that $D'(G)\leq D(G)+1$. For proper colourings, we study relevant invariants called the distinguishing chromatic number $\chi_D(G)$, and the distinguishing chromatic index $\chi'_D(G)$, for vertex and edge colourings, respectively. We show that $\chi_D(G)\leq 2\Delta(G)-1$ for graphs with a finite maximum degree $\Delta(G)$, and we obtain substantially lower bounds for some classes of graphs with infinite motion. We also show that $\chi'_D(G)\leq \chi'(G)+1$, where $\chi'(G)$ is the chromatic index of $G$, and we prove a similar result $\chi''_D(G)\leq \chi''(G)+1$ for proper total colourings. A number of conjectures are formulated.

scholarly journals Vertex Colouring Edge Weightings: a Logarithmic Upper Bound on Weight-Choosability

10.37236/6878 ◽  
2021 ◽  
Vol 28 (2) ◽  
Author(s):  
Kasper Szabo Lyngsie ◽  
Liang Zhong
Keyword(s):  
Upper Bound ◽  
Maximum Degree ◽  
Delta G ◽  
Vertex Colouring ◽  
Total Weighting

A graph $G$ is said to be $(k,m)$-choosable if for any assignment of $k$-element lists $L_v \subset \mathbb{R}$ to the vertices $v \in V(G)$ and any assignment of $m$-element lists $L_e \subset \mathbb{R}$ to the edges $e \in E(G)$  there exists a total weighting $w: V(G) \cup E(G) \rightarrow \mathbb{R}$ of $G$ such that $w(v) \in L_v$ for any vertex $v \in V(G)$ and $w(e) \in L_e$ for any edge $e \in E(G)$ and furthermore, such that for any pair of adjacent vertices $u,v$, we have $w(u)+ \sum_{e \in E(u)}w(e) \neq w(v)+ \sum_{e \in E(v)}w(e)$, where $E(u)$ and $E(v)$ denote the edges incident to $u$ and $v$ respectively. In this paper we give an algorithmic proof showing that any graph $G$ without isolated edges is $(1, 2 \lceil \log_2(\Delta(G)) \rceil+1)$-choosable, where $\Delta(G)$ denotes the maximum degree in $G$.

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