Book contents
- Frontmatter
- Contents
- Preface
- Farewell to Paul Erdős
- Toast to Paul Erdős
- List of Contributors
- Paul Erdős: Some Unsolved Problems
- Menger's Theorem for a Countable Source Set
- On Extremal Set Partitions in Cartesian Product Spaces
- Matchings in Lattice Graphs and Hamming Graphs
- Reconstructing a Graph from its Neighborhood Lists
- Threshold Functions for H-factors
- A Rate for the Erdős–Turán Law
- Deterministic Graph Games and a Probabilistic Intuition
- On Oriented Embedding of the Binary Tree into the Hypercube
- Potential Theory on Distance-Regular Graphs
- On the Length of the Longest Increasing Subsequence in a Random Permutation
- On Richardson's Model on the Hypercube
- Random Permutations: Some Group-Theoretic Aspects
- Ramsey Problems with Bounded Degree Spread
- Hamilton Cycles in Random Regular Digraphs
- On Triangle Contact Graphs
- A Combinatorial Approach to Complexity Theory via Ordinal Hierarchies
- Lattice Points of Cut Cones
- The Growth of Infinite Graphs: Boundedness and Finite Spreading
- Amalgamated Factorizations of Complete Graphs
- Ramsey Size Linear Graphs
- Turán–Ramsey Theorems and Kp-Independence Numbers
- Nearly Equal Distances in the Plane
- Clique Partitions of Chordal Graphs
- On Intersecting Chains in Boolean Algebras
- On the Maximum Number of Triangles in Wheel-Free Graphs
- Blocking Sets in SQS(2v)
- (1,2)-Factorizations of General Eulerian Nearly Regular Graphs
- Oriented Hamilton Cycles in Oriented Graphs
- Minimization Problems for Infinite n-Connected Graphs
- On Universal Threshold Graphs
- Image Partition Regularity of Matrices
- Extremal Graph Problems for Graphs with a Color-Critical Vertex
- A Note on ω1 → ω1 Functions
- Topological Cliques in Graphs
- Local-Global Phenomena in Graphs
- On Random Generation of the Symmetric Group
- On Vertex-Edge-Critically n-Connected Graphs
- On a Conjecture of Erdős and Čudakov
- A Random Recolouring Method for Graphs and Hypergraphs
- Obstructions for the Disk and the Cylinder Embedding Extension Problems
- A Ramsey-Type Theorem in the Plane
- The Enumeration of Self-Avoiding Walks and Domains on a Lattice
- An Extension of Foster's Network Theorem
- Randomised Approximation in the Tutte Plane
- On Crossing Numbers, and some Unsolved Problems
A Random Recolouring Method for Graphs and Hypergraphs
Published online by Cambridge University Press: 06 December 2010
- Frontmatter
- Contents
- Preface
- Farewell to Paul Erdős
- Toast to Paul Erdős
- List of Contributors
- Paul Erdős: Some Unsolved Problems
- Menger's Theorem for a Countable Source Set
- On Extremal Set Partitions in Cartesian Product Spaces
- Matchings in Lattice Graphs and Hamming Graphs
- Reconstructing a Graph from its Neighborhood Lists
- Threshold Functions for H-factors
- A Rate for the Erdős–Turán Law
- Deterministic Graph Games and a Probabilistic Intuition
- On Oriented Embedding of the Binary Tree into the Hypercube
- Potential Theory on Distance-Regular Graphs
- On the Length of the Longest Increasing Subsequence in a Random Permutation
- On Richardson's Model on the Hypercube
- Random Permutations: Some Group-Theoretic Aspects
- Ramsey Problems with Bounded Degree Spread
- Hamilton Cycles in Random Regular Digraphs
- On Triangle Contact Graphs
- A Combinatorial Approach to Complexity Theory via Ordinal Hierarchies
- Lattice Points of Cut Cones
- The Growth of Infinite Graphs: Boundedness and Finite Spreading
- Amalgamated Factorizations of Complete Graphs
- Ramsey Size Linear Graphs
- Turán–Ramsey Theorems and Kp-Independence Numbers
- Nearly Equal Distances in the Plane
- Clique Partitions of Chordal Graphs
- On Intersecting Chains in Boolean Algebras
- On the Maximum Number of Triangles in Wheel-Free Graphs
- Blocking Sets in SQS(2v)
- (1,2)-Factorizations of General Eulerian Nearly Regular Graphs
- Oriented Hamilton Cycles in Oriented Graphs
- Minimization Problems for Infinite n-Connected Graphs
- On Universal Threshold Graphs
- Image Partition Regularity of Matrices
- Extremal Graph Problems for Graphs with a Color-Critical Vertex
- A Note on ω1 → ω1 Functions
- Topological Cliques in Graphs
- Local-Global Phenomena in Graphs
- On Random Generation of the Symmetric Group
- On Vertex-Edge-Critically n-Connected Graphs
- On a Conjecture of Erdős and Čudakov
- A Random Recolouring Method for Graphs and Hypergraphs
- Obstructions for the Disk and the Cylinder Embedding Extension Problems
- A Ramsey-Type Theorem in the Plane
- The Enumeration of Self-Avoiding Walks and Domains on a Lattice
- An Extension of Foster's Network Theorem
- Randomised Approximation in the Tutte Plane
- On Crossing Numbers, and some Unsolved Problems
Summary
We consider a simple randomised algorithm that seeks a weak 2-colouring of a hypergraph H; that is, it tries to 2-colour the points of H so that no edge is monochromatic. If H has a particular well-behaved form of such a colouring, then the method is successful within expected number of iterations O(n3) when H has n points. In particular, when applied to a graph G with n nodes and chromatic number 3, the method yields a 2-colouring of the vertices such that no triangle is monochromatic in expected time O(n4).
A hypergraph H on a set of points V is simply a collection of subsets E of V, the edges of H. A d-graph is a hypergraph in which each edge has size d. A weak 2-colouring of a hypergraph is a partition of the points into two ‘colour’ sets A and B such that each edge E meets both A and B. Deciding if a 3-graph has a weak 2-colouring is NP-complete.
The following simple randomised recolouring method attempts to find a weak 2-colouring of a hypergraph H. It is assumed that we have a subroutine SEEK, which on input of a 2-colouring of the points outputs a monochromatic edge if there is one, and otherwise reports that there are none.
- Type
- Chapter
- Information
- Combinatorics, Geometry and ProbabilityA Tribute to Paul Erdös, pp. 489 - 492Publisher: Cambridge University PressPrint publication year: 1997