Elsevier

Discrete Mathematics

Volume 261, Issues 1–3, 28 January 2003, Pages 377-382
Discrete Mathematics

An application of splittable 4-frames to coloring of Kn,n

Dedicated to Alex Rosa on the occasion of his sixty-five birthday
https://doi.org/10.1016/S0012-365X(02)00483-1Get rights and content
Under an Elsevier user license
open archive

Abstract

Axenovich et al. (J. Combin. Theory Ser. B, to appear) considered the problem of the generalized Ramsey theory. In one case, they use the existence of Steiner triple systems, Pippenger and Spencer's theorem on hyperedge coloring, and the probabilistic method to show that r′(Kn,n,C4,3)⩽3n/4(1+o(1)), where r′(Kn,n,C4,3) denotes the minimum number of colors to color the edges of Kn,n such that every 4-cycle receives at least either 3 colors or 2 alternating colors. In this short paper, using techniques from combinatorial design theory, we prove that r′(Kn,n,C4,3)⩽(2n/3)+9 for all n. The result is the best possible since r′(Kn,n,C4,3)>⌊2n/3⌋ as shown by Axenovich et al. (J. Combin. Theory Ser. B, to appear).

Cited by (0)