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
Clique Partitions of Chordal Graphs
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
To partition the edges of a chordal graph on n vertices into cliques may require as many as n2/6 cliques; there is an example requiring this many, which is also a threshold graph and a split graph. It is unknown whether this many cliques will always suffice. We are able to show that (1 − c)n2/4 cliques will suffice for some c > 0.
Introduction
We consider undirected graphs without loops or multiple edges. The graph Kn on n vertices for which every pair of distinct vertices induces an edge is called a complete graph or a clique on n vertices. If G is any graph, we call any complete subgraph of G a clique of G (we do not require that it be a maximal complete subgraph). A clique covering of G is a set of cliques of G that together contain each edge of G at least once; if each edge is covered exactly once we call it a clique partition. The clique covering number cc(G) and clique partition number cp(G) are the smallest cardinalities of, respectively, a clique covering and a clique partition of G.
The question of calculating these numbers was raised by Orlin in 1977. DeBruijn and Erdős had already proved, in 1948, that partitioning Kn into smaller cliques required at least n cliques. Some more recent studies motivating the current paper include.
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- Combinatorics, Geometry and ProbabilityA Tribute to Paul Erdös, pp. 291 - 298Publisher: Cambridge University PressPrint publication year: 1997