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Clique Partitions of Chordal Graphs†

Published online by Cambridge University Press:  12 September 2008

Paul Erdős ,
Edward T. Ordman  and
Yechezkel Zalcstein
Paul Erdős
Affiliation:
Mathematical Institute, Hungarian Academy of Sciences
Edward T. Ordman
Affiliation:
Memphis State University, Memphis, TN 38152, U.S.A.
Yechezkel Zalcstein
Affiliation:
Division of Computer and Computation Research, National Science Foundation Washington, D.C. 20550, U.S.A.
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Abstract

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.

Type
Research Article
Information
Combinatorics, Probability and Computing , Volume 2 , Issue 4 , December 1993 , pp. 409 - 415
Copyright
Copyright © Cambridge University Press 1993

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References

[1]Bender, E. A., Richmond, L. B. and Wormald, N. C. (1985) Almost all chordal graphs split. J. Austral. Math. Soc. (A) 38 214–221.CrossRefGoogle Scholar
[2]Caccetta, L., Erdős, P., Ordman, E. and Pullman, N. (1985) The difference between the clique numbers of a graph. Ars Combinatoria 19 A 97–106.Google Scholar
[3]Chartrand, G. and Lesniak, L. (1986) Graphs and Digraphs, 2nd. Edition, Wadsworth, Belmont, CA.Google Scholar
[4]Chvátal, V. and Hammer, P. (1973) Set packing and threshold graphs, Univ. of Waterloo Research Report CORR 73–21.Google Scholar
[5]Chvátal, V. and Hammer, P. (1977) Aggregation of inequalities in integer programming. Ann. Discrete Math 1 145–162.CrossRefGoogle Scholar
[6]DeBruijn, N. G. and Erdős, P. (1948) On a combinatorial problem. Indag. Math. 10 421–423.Google Scholar
[7]Erdős, P., Faudree, R. and Ordman, E. (1988) Clique coverings and clique partitions. Discrete Mathematics 72 93–101.CrossRefGoogle Scholar
[8]Erdős, P., Goodman, A. W. and Posa, L. (1966) The representation of a graph by set intersections. Canad. J. Math. 18 106–112.CrossRefGoogle Scholar
[9]Erdős, P., Gyárfás, A., Ordman, E. T. and Zalcstein, Y. (1989) The size of chordal, interval, and threshold subgraphs. Combinatorica 9 (3) 245–253.CrossRefGoogle Scholar
[10]Golumbic, M. (1980) Algorithmic Graph Theory and Perfect Graphs, Academic Press, New York.Google Scholar
[11]Gregory, D. A. and Pullman, N. J. (1982) On a clique covering problem of Orlin. Discrete Math. 41 97–99.CrossRefGoogle Scholar
[12]Henderson, P. and Zalcstein, Y. (1977) A graph theoretic characterization of the PVchunk class of synchronizing primitives. SIAM J. Comp. 6 88–108.CrossRefGoogle Scholar
[13]Orlin, J. (1977) Contentment in Graph Theory: covering graphs with cliques. Indag. Math. 39 406–424.CrossRefGoogle Scholar
[14]Wallis, W. D. (1982) Asymptotic values of clique partition numbers. Combinatorica 2 (1) 99–101.CrossRefGoogle Scholar