Gallai-Ramsey number for K5

被引:5
|
作者
Magnant, Colton [1 ,2 ]
Schiermeyer, Ingo [3 ]
机构
[1] UPS Amer Inc, Supply Chain Solut, 12380 Morris Rd, Atlanta, GA 30005 USA
[2] Acad Plateau Sci & Sustainabil, Xining, Qinghai, Peoples R China
[3] Tech Univ Bergakad Freiberg, Inst Diskrete Math & Algebra, Freiberg, Germany
关键词
complete graphs; Gallai-Ramsey; Ramsey number;
D O I
10.1002/jgt.22835
中图分类号
O1 [数学];
学科分类号
0701 ; 070101 ;
摘要
Given a graph H, the k-colored Gallai-Ramsey number gr(k)(K-3 : H) is defined to be the minimum integer n such that every k-coloring of the edges of the complete graph on n vertices contains either a rainbow triangle or a monochromatic copy of H. Fox et al. conjectured the values of the Gallai-Ramsey numbers for complete graphs. Recently, this conjecture has been verified for the first open case, when H = K-4. In this paper we attack the next case, when H = K-5. Surprisingly it turns out, that the validity of the conjecture depends upon the (yet unknown) value of the Ramsey number R(5, 5). It is known that 43 <= R(5, 5) <= 48 and conjectured that R(5, 5) = 43. If 44 <= R(5, 5) <= 48, then Fox et al.'s conjecture is true and we present a complete proof. If, however, R(5, 5) = 43, then Fox et al.'s conjecture is false, meaning that exactly one of these conjectures is true while the other is false. For the case when R(5, 5) = 43, we show lower and upper bounds for the Gallai-Ramsey number gr(k)(K-3 : K-5).
引用
收藏
页码:455 / 492
页数:38
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