Random Cyclic Triangle-Free Graphs of Prime Order

Yu Jiang,Yanmei Teng,Xiaodong Xu,Meilian Liang,Efthymios G Tsionas

doi:10.1155/2021/5595919

Yu Jiang, Yanmei Teng + Show 3 more

Open Access

https://doi.org/10.1155/2021/5595919

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Abstract

Cyclic triangle-free process (CTFP) is the cyclic analog of the triangle-free process. It begins with an empty graph of order n and generates a cyclic graph of order n by iteratively adding parameters, chosen uniformly at random, subject to the constraint that no triangle is formed in the cyclic graph obtained, until no more parameters can be added. The structure of a cyclic triangle-free graph of the prime order is different from that of composite integer order. Cyclic graphs of prime order have better properties than those of composite number order, which enables generating cyclic triangle-free graphs more efficiently. In this paper, a novel approach to generating cyclic triangle-free graphs of prime order is proposed. Based on the cyclic graphs of prime order, obtained by the CTFP and its variant, many new lower bounds on R 3 , t are computed, including R 3,34 ≥ 230 , R 3,35 ≥ 242 , R 3,36 ≥ 252 , R 3,37 ≥ 264 , R 3,38 ≥ 272 . Our experimental results demonstrate that all those related best known lower bounds, except the bound on R 3,34 , are improved by 5 or more.

Highlights

Ramsey theory [1] has played an important branch in combinatorics, which spans numerous diverse areas of mathematics
In [7], Calkin et al gave theoretical motivation for searching for lower bound for Ramsey numbers based on cyclic graphs of prime order, and provided additional computational evidence that primes tend to perform better than composites. e analysis in [7] does not focus on Ramsey numbers of form R(3, t)
Since the degree of a cyclic triangle-free graph is closely related to its independence number, we study the sizes of parameter sets of cyclic graphs obtained by the Cyclic triangle-free process (CTFP)

Summary

Introduction

Ramsey theory [1] has played an important branch in combinatorics, which spans numerous diverse areas of mathematics. E triangle-free process is an important tool in studying the asymptotic lower bound on R(3, t). The previous work is extended by including an approach to generating cyclic triangle-free graphs of prime order. It is feasible to generate large amount of cyclic triangle-free graphs and improve some previous best known lower bounds on R(3, t), including R(3, 42) ≥ 312, R(3, 44) ≥ 320, R(3, 45) ≥ 337, R(3, 46) ≥ 348, R(3, 47) ≥ 360, R(3, 49) ≥ 376 and.

Preliminaries

The Size of Parameter Set of Cyclic TriangleFree Graphs of Prime Order

Conclusion and Discussions