Vol. 68, No. 1, 1977

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On Ramsey theory and graphical parameters

Linda M. Lesniak and John A. Roberts

Vol. 68 (1977), No. 1, 105–114

A graph G is said to have a factorization into the subgraphs G1,,Gk if the subgraphs are spanning, pairwise edge-disjoint, and the union of their edge sets equals the edge set of G. For a graphical parameter f and positive integers n1,n2,,nk(k 1), the f-Ramsey number f(n1,n2,,nk) is the least positive integer p such that for any factorization Kp = Ik = 1G1, it follows that f(G,) n, for at least one i, 1 i k. In the following, we present two results involving f-Ramsey numbers which hold for various vertex and edge partition parameters, respectively. It is then shown that the concept of f-Ramsey number can be generalized to more than one vertex partition parameter, more than one edge partition parameter, and combinations of vertex and edge partition parameters. Formulas are presented for these generalized f-Ramsey numbers and specific illustrations are given.

Mathematical Subject Classification 2000
Primary: 05C35
Received: 9 June 1975
Revised: 23 July 1976
Published: 1 January 1977
Linda M. Lesniak
John A. Roberts