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Decompositions and Factorizations of Complete Graphs

Chapter

Abstract

Graph decompositions into isomorphic copies of a given graph are a well-established topic studied in both graph theory and design theory. Although spanning tree factorizations may seem to be just a special case of this concept, not many general results are known. We investigate necessary and sufficient conditions for a graph factorization into isomorphic spanning trees to exist.

Keywords

Graph decomposition Graph factorization Graph labeling 

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Notes

Acknowledgments

Supported by the Ministry of Education of the Czech Republic Grant No. MSM6198910027. The author wants to thank the anonymous referees, whose comments helped in improving the quality of this chapter.

References

  1. 1.
    Alspach B, Gavlas H (2001) Cycle decompositions of K n and K nI. J Combin Theory B 81:77–99MATHMathSciNetCrossRefGoogle Scholar
  2. 2.
    Bloom GS (1979) A chronology of the Ringel-Kotzig conjecture and the continuing quest to call all trees graceful. In: Topics in graph theory. The New York Academy of SciencesGoogle Scholar
  3. 3.
    Bosák J (1986) Decompositions of graphs. VEDA, BratislavaGoogle Scholar
  4. 4.
    Bosák J, Erdös P, Rosa A (1972) Decompositions of complete graphs into factors with diameter two. Mat časopis Slov akad vied 22:14–28Google Scholar
  5. 5.
    Bosák J, Rosa A, Znám Š (1968) On decompositions of complete graphs into factors with given diameters. In: Theory of Graphs (Proc Colloq Tihany 1966), Akademiai Kiadó, Budapest, pp 37–56Google Scholar
  6. 6.
    Colbourn CJ, Dinitz JH (eds) (2007) The CRC handbook of combinatorial designs, 2nd edn. Discrete mathematics and its applications. CRC press, Boca RatonGoogle Scholar
  7. 7.
    Diestel R (2006) Graph theory, 3rd edn. Springer, Berlin HeidelbergGoogle Scholar
  8. 8.
    Eldergill P (1997) Decompositions of the complete graph with an even number of vertices. M.Sc. thesis, McMaster University, HamiltonGoogle Scholar
  9. 9.
    El-Zanati S, Vanden Eynden C (1999) Factorizations of K m, n into spanning trees. Graph Combinator 15:287–293MATHMathSciNetCrossRefGoogle Scholar
  10. 10.
    Fronček D (2004) Cyclic decompositions of complete graphs into spanning trees. Discuss Math Graph Theory 24(2):345–353MATHMathSciNetGoogle Scholar
  11. 11.
    Fronček D (2006) Note on factorizations of complete graphs into caterpillars with small diameters. JCMCC 57:179–186MATHGoogle Scholar
  12. 12.
    Fronček D (2007) Bi-cyclic decompositions of complete graphs into spanning trees. Discrete Math 307:1317–1322MATHMathSciNetCrossRefGoogle Scholar
  13. 13.
    Fronček D, Kovářová T (2007) 2n-cyclic blended labeling of graphs. Ars Combinatoria 83:129–144MATHMathSciNetGoogle Scholar
  14. 14.
    Fronček D, Kovář P, Kovářová T, Kubesa M (2010) Factorizations of complete graphs into caterpillars of diameter 5. Discrete Math 310:537–556MATHMathSciNetCrossRefGoogle Scholar
  15. 15.
    Fronček D, Kovář P, Kubesa M Factorizations of complete graphs into trees with at most four non-leave vertices, (submitted)Google Scholar
  16. 16.
    Fronček D, Kubesa M (2002) Factorizations of complete graphs into spanning trees. Congr Numer 154:125–134MATHMathSciNetGoogle Scholar
  17. 17.
    Gallian JA (2003) A dynamic survey of graph labeling. Electron J Combinator DS6Google Scholar
  18. 18.
    Golomb SW (1972) How to number a graph, in graph theory and computing. In: Read RC (ed) Academic, New York, pp 23–37Google Scholar
  19. 19.
    Gronau H-DOF, Grütmüller M, Hartman S, Leck U, Leck V (2002) On orthogonal double covers of graphs. Design Code Cryptogr 27:49–91MATHCrossRefGoogle Scholar
  20. 20.
    Kovářová T (2005) Decompositions of complete graphs into isomorphic spanning trees with given diameters. JCMCC 54:67–81MATHGoogle Scholar
  21. 21.
    Kovářová T (2004) Fixing labelings and factorizations of complete graphs into caterpillars with diameter four. Congr Numer 168:33–48MATHMathSciNetGoogle Scholar
  22. 22.
    Kovářová T (2004) Spanning tree factorizations of complete graphs. Ph.D. thesis, VŠB – Technical University of OstravaGoogle Scholar
  23. 23.
    Kubesa M (2004) Factorizations of complete graphs into [n,r,s,2]-caterpillars of diameter 5 with maximum center. AKCE Int J Graph Combinator 1(2):135–147MATHMathSciNetGoogle Scholar
  24. 24.
    Kubesa M (2004) Factorizations of complete graphs into caterpillars of diameter four and five. Ph.D. thesis, VŠB – Technical University of OstravaGoogle Scholar
  25. 25.
    Kubesa M (2005) Spanning tree factorizations of complete graphs. JCMCC 52:33–49MATHMathSciNetGoogle Scholar
  26. 26.
    Kubesa M (2005) Factorizations of complete graphs into [r,s,2,2]-caterpillars of diameter 5. JCMCC 54:187–193MATHMathSciNetGoogle Scholar
  27. 27.
    Kubesa M (2006) Factorizations of complete graphs into [n,r,s,2]-caterpillars of diameter 5 with maximum end. AKCE Int J Graph Combinator 3(2):151–161MATHMathSciNetGoogle Scholar
  28. 28.
    Kubesa M (2007) Factorizations of complete graphs into [r,s,t,2]-caterpillars of diameter 5. JCMCC 60:181–201MATHMathSciNetGoogle Scholar
  29. 29.
    Meszka M (2008) Solution to the problem of Kubesa. Discuss Math Graph Theory 28:375–378MATHMathSciNetGoogle Scholar
  30. 30.
    Palumbíny D (1973) On decompositions of complete graphs into factors with equal diameters. Boll Un Mat Ital 7:420–428MATHMathSciNetGoogle Scholar
  31. 31.
    Rosa A (1967) On certain valuations of the vertices of a graph. In: Theory of graphs (Intl Symp Rome 1966), Gordon and Breach, Dunod, Paris, pp 349–355Google Scholar
  32. 32.
    Shibata Y, Seki Y (1992) The isomorphic factorization of complete bipartite graphs into trees. Ars Combinatoria 33:3–25MATHMathSciNetGoogle Scholar
  33. 33.
    Tan ND (2008) On a problem of Fronček and Kubesa. Australas J Combinator 40:237–245MATHGoogle Scholar
  34. 34.
    Vetrík T (2006) On factorization of complete graphs into isomorphic caterpillars of diameter 6. Magia 22:17–21Google Scholar
  35. 35.
    West DB (2001) Introduction to graph theory, 2nd edn. Prentice-Hall, Upper Saddle River, NJGoogle Scholar
  36. 36.
    Yap HP (1988) Packing of graphs – a survey. Discrete Math 72:395–404MATHMathSciNetCrossRefGoogle Scholar

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© Springer Science+Business Media, LLC 2011

Authors and Affiliations

  1. 1.Department of Applied MathematicsTechnical University OstravaOstrava–PorubaCzech Republic

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