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 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 Distance Colouring Without One Cycle Length

2018 ◽  
Vol 27 (5) ◽  
pp. 794-807 ◽  
Author(s):  
ROSS J. KANG ◽  
FRANÇOIS PIROT
Keyword(s):  
Maximum Degree ◽  
Cycle Length ◽  
Open Problems ◽  
Logarithmic Factor ◽  
Odd Cycle ◽  
Vertex Colouring ◽  
Edge Colouring

We consider distance colourings in graphs of maximum degree at most d and how excluding one fixed cycle of length ℓ affects the number of colours required as d → ∞. For vertex-colouring and t ⩾ 1, if any two distinct vertices connected by a path of at most t edges are required to be coloured differently, then a reduction by a logarithmic (in d) factor against the trivial bound O(dt) can be obtained by excluding an odd cycle length ℓ ⩾ 3t if t is odd or by excluding an even cycle length ℓ ⩾ 2t + 2. For edge-colouring and t ⩾ 2, if any two distinct edges connected by a path of fewer than t edges are required to be coloured differently, then excluding an even cycle length ℓ ⩾ 2t is sufficient for a logarithmic factor reduction. For t ⩾ 2, neither of the above statements are possible for other parity combinations of ℓ and t. These results can be considered extensions of results due to Johansson (1996) and Mahdian (2000), and are related to open problems of Alon and Mohar (2002) and Kaiser and Kang (2014).

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

10.37236/3153 ◽  
2013 ◽  
Vol 20 (1) ◽  
Author(s):  
Vida Dujmović ◽  
Gwenaël Joret ◽  
Fabrizio Frati ◽  
David R. Wood
Keyword(s):  
Chromatic Number ◽  
Upper Bound ◽  
Planar Graphs ◽  
Open Problems ◽  
Minimum Integer ◽  
Vertex Colouring

A vertex colouring of a graph is nonrepetitive if there is no path for which the first half of the path is assigned the same sequence of colours as the second half. The nonrepetitive chromatic number of a graph $G$ is the minimum integer $k$ such that $G$ has a nonrepetitive $k$-colouring. Whether planar graphs have bounded nonrepetitive chromatic number is one of the most important open problems in the field. Despite this, the best known upper bound is $O(\sqrt{n})$ for $n$-vertex planar graphs. We prove a $O(\log n)$ upper bound.

Topics in Quaternion Linear Algebra

2014 ◽  
Author(s):  
Leiba Rodman
Keyword(s):  
Linear Algebra ◽  
Complex Analysis ◽  
Dimensional Space ◽  
Applied Mathematics ◽  
Three Dimensional ◽  
Number System ◽  
Graduate Course ◽  
Open Problems ◽  
Unpublished Research ◽  
Reference Tool

Quaternions are a number system that has become increasingly useful for representing the rotations of objects in three-dimensional space and has important applications in theoretical and applied mathematics, physics, computer science, and engineering. This is the first book to provide a systematic, accessible, and self-contained exposition of quaternion linear algebra. It features previously unpublished research results with complete proofs and many open problems at various levels, as well as more than 200 exercises to facilitate use by students and instructors. Applications presented in the book include numerical ranges, invariant semidefinite subspaces, differential equations with symmetries, and matrix equations. Designed for researchers and students across a variety of disciplines, the book can be read by anyone with a background in linear algebra, rudimentary complex analysis, and some multivariable calculus. Instructors will find it useful as a complementary text for undergraduate linear algebra courses or as a basis for a graduate course in linear algebra. The open problems can serve as research projects for undergraduates, topics for graduate students, or problems to be tackled by professional research mathematicians. The book is also an invaluable reference tool for researchers in fields where techniques based on quaternion analysis are used.

Continuous Glucose Monitoring Time Series and Hypo/Hyperglycemia Prevention: Requirements, Methods, Open Problems

2008 ◽  
Vol 4 (3) ◽  
pp. 181-192 ◽  
Author(s):  
Giovanni Sparacino ◽  
Andrea Facchinetti ◽  
Alberto Maran ◽  
Claudio Cobelli
Keyword(s):  
Time Series ◽  
Continuous Glucose Monitoring ◽  
Glucose Monitoring ◽  
Open Problems ◽  
Monitoring Time

Secure Cloud Quantum Computing with Verification Based on Quantum Interactive Proof

Impact ◽  
2019 ◽  
Vol 2019 (10) ◽  
pp. 30-32
Author(s):  
Tomoyuki Morimae
Keyword(s):  
Quantum Computing ◽  
Quantum Information ◽  
Theoretical Physics ◽  
View Point ◽  
Research Subjects ◽  
Interactive Proof ◽  
Open Problems ◽  
Main Research ◽  
Proof Systems ◽  
Information Group

In cloud quantum computing, a classical client delegate quantum computing to a remote quantum server. An important property of cloud quantum computing is the verifiability: the client can check the integrity of the server. Whether such a classical verification of quantum computing is possible or not is one of the most important open problems in quantum computing. We tackle this problem from the view point of quantum interactive proof systems. Dr Tomoyuki Morimae is part of the Quantum Information Group at the Yukawa Institute for Theoretical Physics at Kyoto University, Japan. He leads a team which is concerned with two main research subjects: quantum supremacy and the verification of quantum computing.

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