Improved Upper Bounds for Gallai-Ramsey Numbers of Paths and Cycles

Abstract

Given a graph G and a positive integer k, define the Gallai–Ramsey number to be the minimum number of vertices n such that any k‐edge coloring of Kn contains either a rainbow (all different colored) triangle or a monochromatic copy of G. In this work, we improve upon known upper bounds on the Gallai–Ramsey numbers for paths and cycles. All these upper bounds now have the best possible order of magnitude as functions of k.

Gallai-Ramsey number of an 8-cycle

“…Our proof of Theorem 8, particularly the use of Lemma 1 below, suggests that if the Gallai-Ramsey numbers were completely understood for all linear forests, then we may be able to establish the numbers for all cycles. This is somewhat complementary to the results of [10] where the bounds for even cycles were used to establish bounds for paths.…”

Gallai-Ramsey numbers for rainbow S+3 and monochromatic paths

“…Conjecture 4 has been studied by Chung and Graham [3] for t = 3, Liu et al [19] for t = 4, Magnant and Schiermeyer [20] for t = 5. There are also many results for rainbow triangle and monochromatic cycles or paths (see [1,5,8,14,17] and two surveys [10,11]). However, there are not much known about Gallai-Ramsey numbers for other rainbow subgraphs.…”

“…Faudree et al [5] provided a lower bound and Hall et al [17] provided an upper bound for gr k (K 3 : P n ), respectively. Theorem 6 [5,17]. For integers k ≥ 1 and n ≥ 3,…”

Rainbow generalizations of Ramsey theory - a dynamic survey

“…In [68], Faudree, Gould, Jacobson and Magnant proved the following specific Gallai Ramsey numbers Theorem 120 ([68], [100]) Given integers n ≥ 3 and k ≥ 1,…”