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A Degree Sequence Variant of Graph Ramsey Numbers

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Abstract

A (finite) sequence of nonnegative integers is graphic if it is the degree sequence of some simple graph G. Given graphs G 1 and G 2, we consider the smallest integer k such that for every k-term graphic sequence π, there is some graph G with degree sequence π with \({G_1 \subseteq G}\) or with \({G_2 \subseteq \overline{G}}\) . When the phrase “some graph” in the prior sentence is replaced with “all graphs”, the smallest such integer k is the classical Ramsey number r(G 1, G 2). Thus we call this parameter for degree sequences the potential-Ramsey number and denote it r pot (G 1, G 2). In this paper, we give exact values for r pot (K n , K t ), r pot (C n , K t ), and r pot (P n ,K t ) and consider situations where r pot (G 1,G 2) =  r(G 1,G 2).

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References

  1. Busch A., Ferrara M., Hartke S., Jacobson M., Kaul H., West D.: Packing of graphic n-tuples. J. Graph Theory. 70, 29–39 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  2. Dvorak, Z., Mohar, B.: Chromatic number and complete graph substructures for degree sequences, to appear in Combinatorica. Arxiv preprint arXiv:09071583

  3. Erdős, P., Gallai T.: Graphs with prescribed degrees of vertices (hungarian). Mat Lapok. 11, 264–274 (1960)

    Google Scholar 

  4. Erdős P, Faudree R, Rousseau C, Schelp R: On cycle-complete graph Ramsey numbers. J. Graph Theory. 2, 53–64 (1978)

    Article  MathSciNet  Google Scholar 

  5. Erdös, P., Jacobson, M., Lehel, J.: Graphs realizing the same degree sequences and their respective clique numbers. In: Graph Theory, Combinatorics, and Applications, vol. 1, pp. 439–449 (Kalamazoo, 1988). Wiley-Interscience Publications, Wiley

  6. Faudree, R., Schelp, R.: Some problems in Ramsey theory. In: Proceedings of the International Conference on Theory and Applications of Graphs. Western Michigan University, Kalamazoo, 1976. Lecture Notes in Mathematics, vol. 642, pp. 500–515. Springer, Berlin (1978)

  7. Ferrara M., LeSaulnier T., Moffatt C., Wenger P.: On the sum necessary to ensure a degree sequence is potentially h-graphic. Arxiv preprint arXiv:12034611

  8. Ferrara M., Schmitt J.: A general lower bound for potentially H-graphic sequences. SIAM J. Discrete Math. 23, 517–526 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  9. Fink J., Jacobson M.: On n-domination, n-dependence and forbidden subgraphs. In: Graph Theory with Applications to Algorithms and Computer Science, Kalamazoo, pp. 301–311. Wiley-Interscience Publications, Wiley (1984)

  10. Gerencsér L, Gyárfás A: On Ramsey-type problems. Ann. Univ. Sci. Budapest Eotvos Sect. Math. 10, 167–170 (1967)

    MATH  MathSciNet  Google Scholar 

  11. Hakimi, S., Schmeichel, E.: Graphs and their degree sequences: a survey. In: Alavi Y., Lick D. (eds.) Theory and Applications of Graphs. Lecture Notes in Mathematics, vol. 642, pp. 225–235. Springer, Berlin/Heidelberg (1978)

  12. Hakimi S.: On realizability of a set of integers as degrees of the vertices of a linear graph. I. J. Soc. Ind. Appl. Math. 10, 496–506 (1962)

    Article  MATH  MathSciNet  Google Scholar 

  13. Harary, F.: Recent results on generalized Ramsey theory for graphs. In: Proceedings of the Conference on Graph Theory and Applications, Western Michigan University, Kalamazoo, 1972 (dedicated to the memory of J.W.T. Youngs). Lecture Notes in Mathematics, vol. 303, pp. 125–138. Springer, Berlin (1972)

  14. Havel V.: Eine Bemerkung über die Existenz der endlichen Graphen. v Casopis Pv est Mat. 80, 477–480 (2005)

    MathSciNet  Google Scholar 

  15. Li J, Song Z: The smallest degree sum that yields potentially P k -graphical sequences. J. Graph Theory. 29, 63–72 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  16. Luo R.: On potentially C k -graphic sequences. Ars. Combin. 64, 301–318 (2002)

    MATH  MathSciNet  Google Scholar 

  17. Nikiforov V.: The cycle-complete graph Ramsey numbers. Combin. Probab. Comput. 14, 349–370 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  18. Parsons, T.: The Ramsey numbers r(P m , K n ). Discrete Math. 6, 159–162 (1973)

  19. Radziszowski, S.: Small Ramsey numbers. Electron J. Combin. Dyn. Surv. 1, 30 (1994)

    Google Scholar 

  20. Rao, A.R.: The clique number of a graph with a given degree sequence. In: Proceedings of the Symposium on Graph Theory (Indian Statistical Institute, Calcutta, 1976). ISI Lecture Notes, vol. 4, pp. 251–267. Macmillan of India, New Delhi (1979)

  21. Rao, S.B.: A survey of the theory of potentially P-graphic and forcibly P-graphic degree sequences. In: Combinatorics and Graph Theory (Calcutta, 1980). Lecture Notes in Mathematics, vol. 885, pp. 417–440. Springer, Berlin (1981)

  22. Robertson N., Song Z.: Hadwiger number and chromatic number for near regular degree sequences. J. Graph Theory. 64, 175–183 (2010)

    MATH  MathSciNet  Google Scholar 

  23. Yin J.H., Li J.S.: Two sufficient conditions for a graphic sequence to have a realization with prescribed clique size. Discrete Math. 301, 218–227 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  24. Yin, J.H., Li, J.S.: A variation of a conjecture due to Erd H os and Sos. Acta Math. Sin. (Engl Ser) 25, 795–802 (2009)

    Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, University of Dayton, Dayton, USA

    Arthur Busch

  2. Department of Mathematical and Statistical Sciences, University of Colorado Denver, Denver, USA

    Michael Ferrara & Michael Jacobson

  3. Department of Mathematics, University of Nebraska Lincoln, Lincoln, USA

    Stephen G. Hartke

Authors
  1. Arthur Busch
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  2. Michael Ferrara
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  3. Stephen G. Hartke
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  4. Michael Jacobson
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Correspondence to Michael Ferrara.

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Busch, A., Ferrara, M., Hartke, S.G. et al. A Degree Sequence Variant of Graph Ramsey Numbers. Graphs and Combinatorics 30, 847–859 (2014). https://doi.org/10.1007/s00373-013-1307-y

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  • DOI: https://doi.org/10.1007/s00373-013-1307-y

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Mathematics Subject Classification (2000)