On graphical partitions

Abstract

An integer partition {λ12,...,λ v } is said to be graphical if there exists a graph with degree sequence 〈λ i 〉. We give some results corcerning the problem of deciding whether or not almost all partitions of even integer are non-graphical. We also give asymptotic estimates for the number of partitions with given rank.

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References

  1. [1]

    G. Andrews: Sieves for theorems of Euler, Ramanujan and Rogers, in:The Theory of Arithmetic Functions, Lecture Notes in Math.251, (AA. Gioia and D. L. Goldsmith, eds.), 1–20, Springer, Berlin, 1971.

    Google Scholar 

  2. [2]

    A. O. L. Atkin: A note on ranks and conjugacy of partitions,Quart. J. Math. Oxford Ser (2),17 (1966), 335–338.

    Google Scholar 

  3. [3]

    F. C. Auluck, S. Chowla andH. Gupta:J. Indian Math. Soc. 6 (1942) 105–12.

    Google Scholar 

  4. [4]

    D. Bressoud: Extension of the partition Sieve,J. Number Th. 12 (1980), 87–100.

    Google Scholar 

  5. [5]

    F. J. Dyson: Some guesses in the theory of partitions,Eureka (Cambridge),8 (1944), 10–15.

    Google Scholar 

  6. [6]

    P. Erdős andT. Gallai: Graphs with prescribed degrees of vertices, (Hungarian),Mat. Lapok 11 (1960), 264–74.

    Google Scholar 

  7. [7]

    P. Erdős andJ. Lehner: The distribution of the number of summands in the partitions of a positive integer,Duke Math. J. 8 (1941), 335–45.

    Google Scholar 

  8. [8]

    F. Harary:Graph Theory, Addison-Wesley, 1969.

  9. [9]

    C. St. J. A. Nash-Williams: Valency sequences which force graphs to have Hamiltonian circuits; Interim Report C. & O. Research Report, Fac. of Math., University of Waterloo.

  10. [10]

    K. F. Roth andSzekeres: Some asymptotic formulas in the theory of partitions,Quart. J. Math. Oxford Ser. (2) 5 (1954), 241–259.

    Google Scholar 

  11. [11]

    M. Szalay andP. Turán: On some problems of a statistical theory of partitions with application to characters of the symmetric group I,Acta Math. Acad. Scien. Hungaricae 29 (1977), 361–379.

    Google Scholar 

  12. [12]

    M. Szalay andP. Turán: On some problems of a statistical theory of partitions with application to characters of the symmetric group II,Acta Math. Acad. Scien. Hungaricae 29 (1977), 381–392.

    Google Scholar 

  13. [13]

    M. Szalay andP. Turán: On some problems of a statistical theory of partitions with application to characters of the symmetric group III,Acta Math. Acad. Scien. Hungaricae 32 (1978), 129–155.

    Google Scholar 

  14. [14]

    E. M. Wright: The evolution of unlabelled graphs,J. London Math. Soc. 14 (1976), 554–558.

    Google Scholar 

  15. [15]

    E. M. Wright: Graphs on unlabelled nodes with a large number of edges,Proc. London Math. Soc. 28 (1974), 577–94.

    Google Scholar 

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Erdős, P., Richmond, L.B. On graphical partitions. Combinatorica 13, 57–63 (1993). https://doi.org/10.1007/BF01202789

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AMS subject classification code (1991)

  • 10 J 20
  • 05 C 99