Advertisement

On the Nonexistence of Some Generalized Folkman Numbers

  • Xiaodong Xu
  • Meilian Liang
  • Stanisław Radziszowski
Original Paper
  • 7 Downloads

Abstract

For an undirected simple graph G, we write \(G \rightarrow (H_1, H_2)^v\) if and only if for everyred-blue coloring of its vertices there exists a red \(H_1\) or a blue \(H_2\). Thegeneralized vertex Folkman number \(F_v(H_1, H_2; H)\) is defined as the smallest integer n for which there exists an H-free graph G of order n such that \(G \rightarrow (H_1, H_2)^v\). The generalized edge Folkman numbers \(F_e(H_1, H_2; H)\) are defined similarly, when colorings of the edges are considered. We show that \(F_e(K_{k+1},K_{k+1};K_{k+2}-e)\) and \(F_v(K_k,K_k;K_{k+1}-e)\) are well defined for \(k \ge 3\). We prove the nonexistence of \(F_e(K_3,K_3;H)\) for some H, in particular for \(H=B_3\), where \(B_k\) is the book graph of k triangular pages, and for \(H=K_1+P_4\). We pose three problems ongeneralized Folkman numbers, including the existence question of edge Folkmannumbers \(F_e(K_3, K_3; B_4)\), \(F_e(K_3, K_3; K_1+C_4)\) and \(F_e(K_3, K_3; \overline{P_2 \cup P_3} )\). Our results lead to some general inequalities involving two-color and multicolor Folkmannumbers.

Keywords

Folkman number Ramsey number 

Mathematics Subject Classification

05C55 05C35 

References

  1. 1.
    Bikov, A., Nenov, N.: The edge Folkman number \(F_e(3,3;4)\) is greater than 19. Geombinatorics XXVII, 5–14 (2017)MATHGoogle Scholar
  2. 2.
    Dudek, A., Rödl, V.: An almost quadratic bound on vertex Folkman numbers. J. Comb. Theory Ser. B 100, 132–140 (2010)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Dudek, A., Rödl, V.: On the function of Erdős and Rogers (survey). In: Soifer, A. (ed.) Ramsey Theory: Yesterday, Today and Tomorrow. Progress in Mathematics, vol. 285, pp. 63–76. Springer, Berlin (2010)Google Scholar
  4. 4.
    Dudek, A., Rödl, V.: On \(K_s\)-free subgraphs in \(K_{s+k}\)-free graphs and vertex Folkman numbers. Combinatorica 21, 39–53 (2011)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Folkman, J.: Graphs with monochromatic complete subgraphs in every edge coloring. SIAM J. Appl. Math. 18, 19–24 (1970)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Konev, N.: A multiplicative inequality for vertex Folkman numbers. Discret. Math. 308, 4263–4266 (2008)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Lange, A., Radziszowski, S., Xu, X.: Use of MAX-CUT for Ramsey arrowing of triangles. J. Comb. Math. Comb. Comput. 88, 61–71 (2014)MathSciNetMATHGoogle Scholar
  8. 8.
    Lu, L.: Explicit construction of small Folkman graphs. SIAM J. Discret. Math. 21, 1053–1060 (2008)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Nenov, N.: On the small graphs with chromatic number 5 without 4 cliques. Discret. Math. 188, 297–298 (1998)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Nenov, N.: A generalization of a result of Dirac. Annu. Univ. Sofia Fac. Math. Inform. 95, 59–69 (2004)MathSciNetMATHGoogle Scholar
  11. 11.
    Nenov, N.: On the vertex Folkman numbers \(F_v(2,\ldots,2; q)\). Serdica Math. J. 35(3), 251–271 (2009)MathSciNetMATHGoogle Scholar
  12. 12.
    Nešetřil, J., Rödl, V.: The Ramsey property for graphs with forbidden complete subgraphs. J. Comb. Theory Ser. B 20, 243–249 (1976)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Nešetřil, J., Rödl, V.: Simple proof of the existence of restricted Ramsey graphs by means of a partite construction. Combinatorica 1(2), 199–202 (1981)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Radziszowski, S.: Small Ramsey numbers. Electron. J. Comb. Dyn. Surv. DS1, 104, revision #15 (2017). http://www.combinatorics.org
  15. 15.
    Xu, X., Luo, H., Shao, Z.: Upper and lower bounds for \(F_v(4,4;5)\). Electron. J. Comb. N34, 17, 8 (2010). http://www.combinatorics.org
  16. 16.
    Xu, X., Radziszowski, S.: On some open questions for Ramsey and Folkman numbers. In: Ralucca, G., Stephen, H., Craig, L. (eds.) Graph Theory, Favorite Conjectures and Open Problems. Problem Books in Mathematics, vol. 1, pp. 43–62. Springer, Berlin (2016)Google Scholar
  17. 17.
    Xiaodong, X., Shao, Z.: On the lower bound for \(F_v(k, k;k+1)\) and \(F_e(3,4;5)\). Util. Math. 81, 187–192 (2010)MathSciNetMATHGoogle Scholar

Copyright information

© Springer Japan KK, part of Springer Nature 2018

Authors and Affiliations

  • Xiaodong Xu
    • 1
  • Meilian Liang
    • 2
  • Stanisław Radziszowski
    • 3
  1. 1.Guangxi Academy of SciencesNanningPeople’s Republic of China
  2. 2.School of Mathematics and Information ScienceGuangxi UniversityNanningPeople’s Republic of China
  3. 3.Department of Computer ScienceRochester Institute of TechnologyRochesterUSA

Personalised recommendations