Gallai–Ramsey numbers for books

Zou, Jinyu; Mao, Yaping; Magnant, Colton; Wang, Zhao; Ye, Chengfu

doi:10.1016/j.dam.2019.04.013

Cited by 10 publications

(4 citation statements)

References 12 publications

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“…In this paper, we study Gallai-Ramsey numbers for some graphs in H and K m for m ≥ 2. Recently, Gallai-Ramsey numbers for many graphs above have been determined in [15] for H 1 and H 2 , [9] for the graphs from H 3 to H 6 , [16] for H 7 , [14], [17] for H 8 and [12] for H 9 . However, the Gallai-Ramsey numbers for the remaining graphs H 10 , H 11 and H 12 have not been determined so far.…”

Section: Theorem 12 ([6]mentioning

confidence: 99%

Gallai-Ramsey numbers for graphs with chromatic number three

Zhao¹,

Wei²

2020

Preprint

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Given a graph H and an integer k ≥ 1, the Gallai-Ramsey number GR k (H) is defined to be the minimum integer n such that every k-edge coloring of K n contains either a rainbow (all different colored) triangle or a monochromatic copy of H. In this paper, we study Gallai-Ramsey numbers for graphs with chromatic number three such as K m for m ≥ 2, where K m is a kipas with m+1 vertices obtained from the join of K 1 and P m , and a class of graphs with five vertices, denoted by H . We first study the general lower bound of such graphs and propose a conjecture for the exact value of GR k ( K m ). Then we give a unified proof to determine the Gallai-Ramsey numbers for many graphs in H and obtain the exact value of GR k ( K 4 ) for k ≥ 1. Our outcomes not only indicate that the conjecture on GR k ( K m ) is true for m ≤ 4, but also imply several results on GR k (H) for some H ∈ H which are proved individually in different papers.

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Section: Theorem 12 ([6]mentioning

confidence: 99%

Gallai-Ramsey numbers for graphs with chromatic number three

Zhao¹,

Wei²

2020

Preprint

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“…The Gallai-Ramsey numbers gr k (K 3 : H) for all the graphs H on five vertices and at most seven edges are obtained (see [10,17,14,16]). There are two graphs on five vertices and eight edges, one of which is a wheel graph W 4 , and the other is a graph B + 3 obtained from the book graph B 3 by adding an edge between two vertices with degree two.…”

Section: Introductionmentioning

confidence: 99%

Gallai-Ramsey numbers for graphs with five vertices and eight edges

Su,

Liu

2020

Preprint

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A Gallai k-coloring is a k-edge coloring of a complete graph in which there are no rainbow triangles. For given graphs G 1 , G 2 , G 3 and nonnegative integers r, s, t) is the minimum integer n such that every Gallai k-colored K n contains a monochromatic copy of G 1 colored by one of the first r colors or a monochromatic copy of G 2 colored by one of the middle s colors or a monochromatic copy of G 3 colored by one of the last t colors. In this paper, we determine the value of Gallai-Ramsey number in the case that G 1 = B + 3 , G 2 = S + 3 and G 3 = K 3 . Then the Gallai-Ramsey number gr k (K 3 : B + 3 ) is obtained. Thus the Gllai-Ramsey numbers for graphs with five vertices and eight edges are solved completely. Furthermore, the the Gallai-Ramsey numbers gr k (K 3 : r) and gr k (K 3 : s • S + 3 , (k − s) • K 3 ) are obtained, respecticely.

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“…It is known that there are twenty three isolated-free graphs with five vertices, denoted by H . The Gallai-Ramsey numbers for the graphs in H with chromatic number two and three were completely determined in [3,11,14,15,[18][19][20][21][22][23]. In this paper, we obtain the Gallai-Ramsey numbers for graphs in H with chromatic number four as follows.…”

Section: Introductionmentioning

confidence: 99%