Vol. 12, No. 5, 2019

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On some edge Folkman numbers, small and large

Jenny M. Kaufmann, Henry J. Wickus and Stanisław P. Radziszowski

Vol. 12 (2019), No. 5, 813–822
Abstract

Edge Folkman numbers Fe(G1,G2;k) can be viewed as a generalization of more commonly studied Ramsey numbers. Fe(G1,G2;k) is defined as the smallest order of any Kk-free graph F such that any red-blue coloring of the edges of F contains either a red G1 or a blue G2. In this note, first we discuss edge Folkman numbers involving graphs Js = Ks e, including the results Fe(J3,Kn;n + 1) = 2n 1, Fe(J3,Jn;n) = 2n 1, and Fe(J3,Jn;n + 1) = 2n 3. Our modification of computational methods used previously in the study of classical Folkman numbers is applied to obtain upper bounds on Fe(J4,J4;k) for all k > 4.

Keywords
Folkman numbers, Ramsey numbers
Mathematical Subject Classification 2010
Primary: 05C55
Milestones
Received: 3 June 2018
Revised: 23 October 2018
Accepted: 29 November 2018
Published: 22 May 2019

Communicated by Kenneth S. Berenhaut
Authors
Jenny M. Kaufmann
Department of Mathematics
Princeton University
Princeton, NJ
United States
Henry J. Wickus
Department of Mathematics and Computer Science
DeSales University
Center Valley, PA
United States
Stanisław P. Radziszowski
Department of Computer Science
Rochester Institute of Technology
Rochester, NY
United States