Graphs and Combinatorics

, Volume 34, Issue 5, pp 1101–1110 | Cite as

On the Nonexistence of Some Generalized Folkman Numbers

  • Xiaodong XuEmail author
  • Meilian Liang
  • Stanisław Radziszowski
Original Paper


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.


Folkman number Ramsey number 

Mathematics Subject Classification

05C55 05C35 


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Copyright information

© Springer Japan KK, part of Springer Nature 2018

Authors and Affiliations

  • Xiaodong Xu
    • 1
    Email author
  • 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

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