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Clique coverings of graphs V: maximal-clique partitions

Published online by Cambridge University Press:  17 April 2009

N.J. Pullman
Affiliation:
Department of Mathematics and Statistics, Queen's University, Kingston, Ontario K7L 3N6, Canada
H. Shank
Affiliation:
Department of Mathematics and Statistics, Queen's University, Kingston, Ontario K7L 3N6, Canada
W.D. Wallis
Affiliation:
Department of Mathematics, University of Newcastle, Newcastle, New South Wales 2308, Australia.
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Abstract

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A maximal-clique partition of a graph G is a way of covering G with maximal complete subgraphs, such that every edge belongs to exactly one of the subgraphs. If G has a maximal-clique partition, the maximal-clique partition number of G is the smallest cardinality of such partitions. In this paper the existence of maximal-clique partitions is discussed – for example, we explicitly describe all graphs with maximal degree at most four which have maximal-clique partitions - and discuss the maximal-clique partition number and its relationship to other clique covering and partition numbers. The number of different maximal-clique partitions of a given graph is also discussed. Several open problems are presented.

Type
Research Article
Copyright
Copyright © Australian Mathematical Society 1982

References

[1]Donald, Alan, “Edge and arc partitions of arbitrary graphs and digraphs” (MSc Thesis, Queen's University, Kingston, Canada, 1979).Google Scholar
[2]Erdös, Paul, Goodman, A.W., and Pósa, Louis, “The representation of a graph by set intersections”, Canad. J. Math. 18 (1966), 106112.CrossRefGoogle Scholar
[3]Euler, Leonhard, “Recherches sur une nouvelle espèce de quarrés magiques”, Verh. Zeeuusah. Genootsch. Wetensch. Vlissingen 9 (1782), 85239.Google Scholar
[4]Fisher, R.A. and Yates, F., “The 6 × 6 Latin squares”, Proc. Cambridge Philos. Soc. 30 (19331934), 492507.CrossRefGoogle Scholar
[5]Hall, Marshall Jr, “Distinct representatives of subsets”, Bull. Amer. Math. Soc. 54 (1948), 922926.CrossRefGoogle Scholar
[6]Harary, Frank, Graph theory (Addison-Wesley, Reading, Massachusetts; Menlo Park, California; London; 1969).Google Scholar
[7]Holzmann, Carlos A. and Harary, Frank, “On a tree graph of a matroid”, SIAM J. Appl. Math. 22 (1972), 187193.CrossRefGoogle Scholar
[8]Lovász, L., “on covering of graphs” Theory of graphs, 231236 (Proc. Colloq. Tihany, 1966. Academic Press, New York and London, 1968).Google Scholar
[9]Maurer, Stephen B., “Matroid basis graphs. I“, J. Combin. Theory Ser. B 14 (1973), 216240.CrossRefGoogle Scholar
[10]Orlin, James, “Contentment in graph theory: covering graphs with cliques”, Neder. Akad. Wetensch. Proc. Ser. A 80 = Indag. Math. 39 (1977), 406424.Google Scholar
[11]Pullman, Norman J., “Clique coverings of graphs IV: algorithms”, SIAM J. Comp. (to appear).Google Scholar
[12]Pullman, Norman J. and Caen, Dom de, “Clique coverings of graphs III: clique coverings of regular graphs”, Congr. Numer. 29 (1980), 795808.Google Scholar
[13]Pullman, Norman J. and Caen, Dom de, “Clique coverings of graphs – I: clique partitions of regular graphs”, Utilitas Math. 19 (1981), 177205.Google Scholar
[14]Pullman, Norman J. and Donald, Alan, “Clique coverings of graphs – II: complements of cliques”, Utilitas Math. 19 (1981), 207213.Google Scholar
[15]Ryser, H.J., “Intersection properties of finite sets”, J. Combin. theory Ser. A 14 (1973), 7992.Google Scholar
[16]Ryser, H., “Intersection properties of finite sets”, Colloquio Internazionale sulle Teorie Combinatorie (Rome 1973), II, 321334 (Atti dei Convegni Lincei, No. 17, Accad. Naz. Lincei, Rome, 1976).Google Scholar
[17]Sade, Albert, “Enumération des carrés Latins. Applications au 7e ordre. Conjecture pour les ordres supérieures” (Published by the author, Marseille, 1948).Google Scholar
[18]Vajda, S., Patterns and configurations in finite spaces (Griffin's Statistical Monographs and Courses, 22. Charles Griffin, London; Hafner, New York; 1967).Google Scholar
[19]Wells, Mark B., “The number of Latin squares of order 8”, J. Combin. Theory 3 (1967), 9899.CrossRefGoogle Scholar