New Lower Bounds for Two Multicolor Vertex Folkman Numbers

Shao, Zehui; Liang, Meilian; He, Jifeng; Xu, Xin-Shun

doi:10.1109/caman.2011.5778782

Cited by 4 publications

(4 citation statements)

References 6 publications

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“…At the end of the paper, we consider the vertex Folkman number F v (2, 3, 3; 4) and prove 20 ≤ F v (2, 3, 3; 4) ≤ 24 (Theorem 4.1). This is an improvement over the known bounds 19 ≤ F v (2, 3, 3; 4) ≤ 30 from [25] and [24]. This paper is organized in 4 sections.…”

Section: Introductionmentioning

confidence: 75%

The edge Folkman number $F_e(3, 3; 4)$ is greater than 19

Bikov,

Nenov

2016

Preprint

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The set of the graphs which do not contain the complete graph on q vertices Kq and have the property that in every coloring of their edges in two colors there exist a monochromatic triangle is denoted by He(3, 3; q). The edge Folkman numbers Fe(3, 3; q) = min {| V(G)| : G ∈ He(3, 3; q)} are considered. Folkman proved in 1970 that Fe(3, 3; q) exists if and only if q ≥ 4. From the Ramsey number R(3, 3) = 6 it becomes clear that Fe(3, 3; q) = 6 if q ≥ 7. It is also known that Fe(3, 3; 6) = 8 and Fe(3, 3; 5) = 15. The upper bounds on the number Fe(3, 3; 4) which follow from the construction of Folkman and from the constructions of some other authors are not good. In 1975 Erdos posed the problem to prove the inequality Fe(3, 3; 4) < 10 10 . This Erdos problem was solved by Spencer in 1978. The last upper bound on Fe(3, 3; 4) was obtained in 2012 by Lange, Radziszowski and Xu, who proved that Fe(3, 3; 4) ≤ 786. The best lower bound on this number is 19 and was obtained 10 years ago by Radziszowski and Xu. In this paper, we improve this result by proving Fe(3, 3; 4) ≥ 20. At the end of the paper, we improve the known bounds on the vertex Folkman number Fv(2, 3, 3; 4) by proving 20 ≤ Fv(2, 3, 3; 4) ≤ 24.

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Section: Introductionmentioning

confidence: 75%

The edge Folkman number $F_e(3, 3; 4)$ is greater than 19

Bikov,

Nenov

2016

Preprint

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“…Similar algorithms are used in [2], [30], [9] and [26]. Also, with the help of computer, results for Folkman numbers are obtained in [6], [28], [27] and [3]. The naive approach for finding all graphs in H(2, 2, 5; 6; 16) suggests to check all graphs of order 16 for inclusion in H(2, 2, 5; 6).…”

Section: Proof Of Theorem 14mentioning

confidence: 99%

The vertex Folkman numbers $F_v(a_1, ..., a_s; m - 1) = m + 9$, if $\max\{a_1, ..., a_s\} = 5$

Bikov,

Nenov

2015

Preprint

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For a graph G the expression G v → (a1, ..., as) means that for any scoloring of the vertices of G there exists i ∈ {1, ..., s} such that there is a monochromatic ai-clique of color i. The vertex Folkman numbers Fv(a1, ..., as; m − 1) = min{| V(G)| : G v → (a1, ..., as) and Km−1 ⊆ G}.are considered, where m = s i=1 (ai − 1) + 1. With the help of computer we show that Fv(2, 2, 5; 6) = 16 and then we prove Fv(a1, ..., as; m − 1) = m + 9, if max{a1, ..., as} = 5.We also obtain the bounds m + 9 ≤ Fv(a1, ..., as; m − 1) ≤ m + 10, if max {a1, ..., as} = 6.

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“…Similar algorithms are used in [1], [2], [19], [8], [15]. Also with the help of the computer, results for Folkman numbers are obtained in [6], [17], [16] and [3]. The following proposition for maximal graphs in H(m p ; q; n) will be useful Proposition 3.1.…”

Section: Algorithmsmentioning

confidence: 99%

Modified vertex Folkman numbers

Bikov,

Nenov

2015

Preprint

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Let a 1 , ..., as be positive integers. For a graph G the expressionmeans that for every coloring of the vertices of G in s colors (s-coloring) there exists i ∈ {1, ..., s}, such that there is a monochromatic a i -clique of color i. If m and p are positive integers, thenmeans that for arbitrary positive integers a 1 , ..., as (s is not fixed), such that s i=1The modified vertex Folkman numbers are defined by the equalityIf q ≥ m these numbers are known and they are easy to compute. In the case q = m − 1 we know all of the numbers when p ≤ 5. In this work we consider the next unknown case p = 6 and we prove with the help of a computer that F (m 6 ; m − 1) = m + 10.

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New Lower Bounds for Two Multicolor Vertex Folkman Numbers

Cited by 4 publications

References 6 publications

The edge Folkman number $F_e(3, 3; 4)$ is greater than 19

The edge Folkman number $F_e(3, 3; 4)$ is greater than 19

The vertex Folkman numbers $F_v(a_1, ..., a_s; m - 1) = m + 9$, if $\max\{a_1, ..., a_s\} = 5$

Modified vertex Folkman numbers

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