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Clique Partitions of Chordal Graphs

Published online by Cambridge University Press:  06 December 2010

Paul Erdős
Affiliation:
Mathematical Institute, Hungarian Academy of Sciences
E.T. Ordman
Affiliation:
Memphis State University, Memphis, TN 38152 U.S.A
Y. Zalcstein
Affiliation:
Division of Computer and Computation Research, National Science Foundation, Washington, D. C. 20550, U. S. A
Béla Bollobás
Affiliation:
University of Cambridge
Andrew Thomason
Affiliation:
University of Cambridge
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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.

Type
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Combinatorics, Geometry and Probability
A Tribute to Paul Erdös
, pp. 291 - 298
Publisher: Cambridge University Press
Print publication year: 1997

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  • Clique Partitions of Chordal Graphs
    • By Paul Erdős, Mathematical Institute, Hungarian Academy of Sciences, E.T. Ordman, Memphis State University, Memphis, TN 38152 U.S.A, Y. Zalcstein, Division of Computer and Computation Research, National Science Foundation, Washington, D. C. 20550, U. S. A
  • Edited by Béla Bollobás, University of Cambridge, Andrew Thomason, University of Cambridge
  • Book: Combinatorics, Geometry and Probability
  • Online publication: 06 December 2010
  • Chapter DOI: https://doi.org/10.1017/CBO9780511662034.027
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  • Clique Partitions of Chordal Graphs
    • By Paul Erdős, Mathematical Institute, Hungarian Academy of Sciences, E.T. Ordman, Memphis State University, Memphis, TN 38152 U.S.A, Y. Zalcstein, Division of Computer and Computation Research, National Science Foundation, Washington, D. C. 20550, U. S. A
  • Edited by Béla Bollobás, University of Cambridge, Andrew Thomason, University of Cambridge
  • Book: Combinatorics, Geometry and Probability
  • Online publication: 06 December 2010
  • Chapter DOI: https://doi.org/10.1017/CBO9780511662034.027
Available formats
×

Save book to Google Drive

To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Google Drive.

  • Clique Partitions of Chordal Graphs
    • By Paul Erdős, Mathematical Institute, Hungarian Academy of Sciences, E.T. Ordman, Memphis State University, Memphis, TN 38152 U.S.A, Y. Zalcstein, Division of Computer and Computation Research, National Science Foundation, Washington, D. C. 20550, U. S. A
  • Edited by Béla Bollobás, University of Cambridge, Andrew Thomason, University of Cambridge
  • Book: Combinatorics, Geometry and Probability
  • Online publication: 06 December 2010
  • Chapter DOI: https://doi.org/10.1017/CBO9780511662034.027
Available formats
×