Topics in Gallai-Ramsey Theory

Abstract

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Gallai–Ramsey number for K5 ${K}_{5}$

“…Next we observe that $${F}_{3}$$ is an $$({R}_{3},{B}_{3})$$‐graph. If $$w({F}_{3})\le 4$$, then we obtain $$$w(H)\le \frac{1}{R}(2\cdot 3.25+5\cdot 3.25+4)=\frac{26.75}{R},$$$ which gives a contradiction as before [9]. So we may assume that $$w({F}_{3})\gt 4$$.…”

“…is the minimum number of vertices n such that every k-coloring of the edges of K N for ≥ N n must contain a monochromatic copy of G. We refer to [12] for a dynamic survey of known Ramsey numbers. As a restricted version of the Ramsey number, the k-color Gallai-Ramsey number gr K G ( : ) k 3 is defined to be the minimum integer n such that every k-coloring of the edges of K N for ≥ N n must contain either a rainbow triangle or a monochromatic copy of G. We refer to [3] for a dynamic survey of known Gallai-Ramsey numbers and the recent book [9]. In particular, the following was recently conjectured for complete graphs.…”

Gallai-Ramsey numbers involving a rainbow $4$-path

“…With the additional restriction of forbidding the rainbow copy of G, it is clear that gr k (G : H) ≤ R k (H) for any G. Till now, most work focuses on the case G = K 3 ; see [3,6,8,11,11,15,16,20,21,22,30]. For more details on the Gallai-Ramsey numbers, we refer to the book [19] and a survey paper [7].…”

Star-critical Gallai-Ramsey numbers involving the disjoint union of triangles

“…All of the concepts defined so far have analogues in the setting of Gallai colorings. See [6,7], and [16] for an overview of Gallai-Ramsey numbers. A Gallai t-coloring of a graph G is a t-coloring τ of G that lacks rainbow triangles.…”