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A status on the linear arboricity

  • J. Akiyama
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 108)

Abstract

In a inear forest, each component is a path. The linear arboricity ≡(G) of a graph G is defined in Harary [8] as the minimum number of linear forests whose union is G. This invariant first arose in a study [10] of information retrieval in file systems. A quite similar covering invariant which is well known to the linear arboricity is the arboricity of a graph, which is defined as the minimum number of forests whose union is G. Nash-Williams [11] determined the arboricity of any graph, however only few results on the linear arboricity are known. We shall present these discoveries and an open problem on this new invariant.

References

  1. [1]
    J.Akiyama and V.Chvátal, Another proof of the linear arboricity for cubic graphs, to appear.Google Scholar
  2. [2]
    J.Akiyama, G.Exoo and F.Harary, Covering and packing in graphs III: Cyclic and acyclic invariants. Math. Slovaca 29(1980)Google Scholar
  3. [3]
    J.Akiyama, G.Exoo and F.Harary, Covering and packing in graphs IV: Linear arboricity. Networks 11(1981)Google Scholar
  4. [4]
    J. Akiyama and T. Hamada, The decompositions of line graphs, middle graphs and total graphs of complete graphs into forests. Discrete Math. 26(1979)203–208.Google Scholar
  5. [5]
    J.Akiyama and I.Sato, A comment on the linear arboricity for regular multigraphs, to appear.Google Scholar
  6. [6]
    M. Behzad, G. Chartrand and L. Lesniak-Foster, Graphs and Digraphs, Prindle, Weber & Schmidt, Boston (1979)Google Scholar
  7. [7]
    H.Enomoto, The linear arboricity of cubic graphs and 4-regular graphs, Private communication.Google Scholar
  8. [8]
    F. Harary, Covering and packing I, Ann. N.Y.Acad. Sci. 175(1970)198–205.Google Scholar
  9. [9]
    F.Harary, Graph Theory, Addison-Wesley, Mass. (1969)Google Scholar
  10. [10]
    F. Harary and D. Hsiao, A formal system for information retrieval files, Comm.A.C.M., 13(1970)67–73.Google Scholar
  11. [11]
    C. Nash-Williams, Decomposition of finite graphs into forests. J. London Math Soc. 39(1964)12.Google Scholar
  12. [12]
    B.Peroche, On partition of graphs into linear forests and dissections, Rapport de recherche, Centre National de la recherche scientifique Google Scholar
  13. [13]
    J. Petersen, Die Theorie der regularen Graphen, Acta Math. 15(1891)193–200.Google Scholar
  14. [14]
    R. Stanton, D. Cowan and L. James, Some results on path numbers, Proc. Louisiana Conf. Combinatorics, Graph Theory and Computing, Baton, Rouge (1970)112–135.Google Scholar
  15. [15]
    W. Tutte, The subgraph problem, Advances in Graph Theory (B. Bollbás, ed.) North-Holland, Amsterdam (1978)289–295.Google Scholar
  16. [16]
    V. Vizing, On an estimate of the chromatic class of p-graph, Diskret. Analiz. 3(1964)25–30.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1981

Authors and Affiliations

  • J. Akiyama
    • 1
  1. 1.Department of MathematicsNippon Ika UniversityKawasakiJapan

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