Graph Theory pp 101-113 | Cite as

The Path Partition Conjecture

  • Marietjie Frick
  • Jean E. DunbarEmail author
Part of the Problem Books in Mathematics book series (PBM)


The Path Partition Conjecture (PPC) states that if G is any graph and (a, b) any pair of positive integers such that G has no path with more than a + b vertices, then there exists a partition (A, B) of the vertex set of G such that A has no path with more than a vertices, and B has no path with more than b vertices. We present a brief history of the PPC, discuss its relation to other conjectures and examine results supporting the PPC that have appeared in the literature since its first formulation in 1981. We conclude with a few related open problems.



M. Frick is supported by the National Research Foundation of S.A. Grant 107668.


  1. 1.
    R.E.L. Aldred, C. Thomassen, Graphs with not all possible path-kernels. Discrete Math. 285, 297–300 (2004)MathSciNetzbMATHGoogle Scholar
  2. 2.
    L.W. Beineke, J.E. Dunbar, M. Frick, Detour-saturated graphs. J. Graph Theory 49, 116–134 (2005)MathSciNetzbMATHGoogle Scholar
  3. 3.
    M. Borowiecki, I. Broere, M. Frick, P. Mihók, G. Semanis~in, A survey of hereditary properties of graphs. Discuss. Math. Graph Theory 17, 5–50 (1997)MathSciNetzbMATHGoogle Scholar
  4. 4.
    S. Brandt, R. Faudree, W. Goddard, Weakly pancyclic graphs. J. Graph Theory 27(3), 142–176 (1998)MathSciNetzbMATHGoogle Scholar
  5. 5.
    S. Brandt, O. Favaron, Z. Ryjáček, Closure and stable Hamiltonian properties in claw-free graphs. J. Graph Theory 34(1), 31–41 (2000)MathSciNetzbMATHGoogle Scholar
  6. 6.
    I. Broere, P. Hajnal, P. Mihók, Partition problems and kernels of graphs. Discuss. Math. Graph Theory 17, 311–313 (1997)MathSciNetzbMATHGoogle Scholar
  7. 7.
    I. Broere, M. Dorfling, J.E. Dunbar, M. Frick, A path(ological) partition problem. Discuss. Math. Graph Theory 18, 113–125 (1998)MathSciNetzbMATHGoogle Scholar
  8. 8.
    I. Broere, S. Dorfling, E. Jonck, Generalized chromatic numbers and additive hereditary properties of graphs. Discuss. Math. Graph Theory 22, 259–270 (2002)MathSciNetzbMATHGoogle Scholar
  9. 9.
    F. Bullock, M. Frick, Detour chromatic numbers of graphs. Discuss. Math. Graph Theory 21, 283–291 (2001)MathSciNetzbMATHGoogle Scholar
  10. 10.
    F. Bullock, J.E. Dunbar, M. Frick, Path partitions and P n-free sets. Discrete Math. 289, 145–155 (2004)MathSciNetzbMATHGoogle Scholar
  11. 11.
    G. Chartrand, D.P. Geller, S. Hedetniemi, A generalization of the chromatic number. Proc. Camb. Philos. Soc. 64, 265–271 (1968)MathSciNetzbMATHGoogle Scholar
  12. 12.
    D.G. Corneil, H. Lerchs, L.S. Burlingham, Complement reducible graphs. Discrete Appl. Math. 3, 163–174 (1981)MathSciNetzbMATHGoogle Scholar
  13. 13.
    J.E. Dunbar, M. Frick, Path kernels and partitions. J. Combin. Math. Combin. Comput. 31, 137–149 (1999)MathSciNetzbMATHGoogle Scholar
  14. 14.
    J.E. Dunbar, M. Frick, The path partition conjecture is true for claw-free graphs. Discrete Math. 307, 1285–1290 (2007)MathSciNetzbMATHGoogle Scholar
  15. 15.
    H. Fleischner, In the square of graphs, hamiltonicity and pancyclicity, hamiltonian connectedness and panconnectedness are equivalent concepts. Monatshefte für Mathematik 82, 125–149 (1976)MathSciNetzbMATHGoogle Scholar
  16. 16.
    M. Frick, I. Schiermeyer, An asymptotic result on the path partition conjecture. Electron. J. Combin. 12, R48 (2005)MathSciNetzbMATHGoogle Scholar
  17. 17.
    T. Gallai, On directed paths and circuits, in Theory of Graphs (Academic Press, New York, 1968), pp. 115–118zbMATHGoogle Scholar
  18. 18.
    A.N. Glebov, D.J. Zambalaeva, Path partitioning planar graphs, Siberian. Electronic Mathematical Reports 4, 450–459 (2007). (in Russian, English abstract)
  19. 19.
    W. He, B. Wang, A note on path kernels and partitions. Discrete Math. 310, 3040–3042 (2010)MathSciNetzbMATHGoogle Scholar
  20. 20.
    S. Hedetniemi, My top 10 graph theory conjectures and open problems, in: Graph Theory, ed. by R. Gera, S.T. Hedetniemi, C. Larson. Favorite Conjectures and Open Problems, vol. 1 (Springer, Berlin, 2016)zbMATHGoogle Scholar
  21. 21.
    P. Katrenic~, G. Semanis~in, A note on the path kernel conjecture. Discrete Math. 309, 2551–2554 (2009)Google Scholar
  22. 22.
    T. Kaiser, M. Kriesell, On the pancyclicity of lexicographic products. Graphs Combin. 22, 51–58 (2006)MathSciNetzbMATHGoogle Scholar
  23. 23.
    J.M. Laborde, C. Payan, N.H. Xuong, Independent sets and longest directed paths in digraphs, in Graphs and Other Combinatorial Topics, Prague, 1982, pp. 173–177. Teubner-Texte Math., 59, 1983Google Scholar
  24. 24.
    D.R. Lick, A.T. White, k-degenerate graphs. Can. J. Math. 22, 1082–1096 (1970)MathSciNetzbMATHGoogle Scholar
  25. 25.
    L. Lovász, On decomposition of graphs. Stud. Sci. Math. Hungar. 1, 237–238 (1966)zbMATHGoogle Scholar
  26. 26.
    L.S. Melnikov, I.V. Petrenko, On path kernels and partitions of undirected graphs. Diskretn Anal. Issled Oper. 9, 21–35 (2002) (in Russian)Google Scholar
  27. 27.
    L.S. Melnikov, I.V. Petrenko, Path kernels and partitions of graphs with small cycle length, Methods and tools of program construction and optimization, ed. by V.N. Kasyanov (ISI SB Russian Academy of Science, Novosibirsk, 2005), pp. 145–160 (in Russian)Google Scholar
  28. 28.
    P. Mihók, Problem 4 in Graphs, Hypergraphs and Matroids, ed. by M. Borowiecki, Z. Skupien (Zielona Góra, 1985), p. 86Google Scholar
  29. 29.
    Z. Ryjáček, On a closure concept in claw-free graphs. J. Combin. Theory Series B 70, 217–224 (1997)MathSciNetzbMATHGoogle Scholar
  30. 30.
    J. Vronka, Vertex sets of graphs with prescribed properties (in Slovak), Thesis, supervised by P. Mihók, P.J. Safárik University, Košice, 1986Google Scholar
  31. 31.
    D.B. West, Research problems. Discrete Math. 272, 301–306 (2003)Google Scholar

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Authors and Affiliations

  1. 1.Department of Mathematics and Applied MathematicsUniversity of PretoriaPretoriaSouth Africa
  2. 2.Converse CollegeSpartanburgUSA

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