Random Cyclic Triangle-Free Graphs of Prime Order

Jiang, Yu; Liang, Meilian; Teng, Yanmei; Xu, Xiaodong

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

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Abstract Introduction Preliminaries Conclusion Data Availability Conflicts of Interest Acknowledgments References Copyright Related Articles

Research Article | Open Access

Volume 2021 | Article ID 5595919 | https://doi.org/10.1155/2021/5595919

Random Cyclic Triangle-Free Graphs of Prime Order

Yu Jiang,¹Meilian Liang,²Yanmei Teng,³and Xiaodong Xu⁴

Academic Editor: Efthymios G. Tsionas

Received25 Feb 2021

Accepted28 Jul 2021

Published09 Aug 2021

Abstract

Cyclic triangle-free process (CTFP) is the cyclic analog of the triangle-free process. It begins with an empty graph of order and generates a cyclic graph of order 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 are computed, including , , , , . Our experimental results demonstrate that all those related best known lower bounds, except the bound on , are improved by 5 or more.

1. Introduction

Ramsey theory [1] has played an important branch in combinatorics, which spans numerous diverse areas of mathematics. Many research efforts have been devoted into computing Ramsey numbers and their generalizations [2–5].

Let be a set of integers ( and ), a graph with the vertex set and the edge set is a cyclic graph of order , namely . is the parameter set of . Let and be two positive integers, the Ramsey number for and , denoted by , is the minimum positive integer such that every graph of order contains either an -clique or a -independent set. There are many results and open problems in Ramsey theory in terms of computing lower bounds for Ramsey numbers (see [6]).

The Ramsey number is an important topic in Ramsey theory. 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. The analysis in [7] does not focus on Ramsey numbers of form . For , in [7] it was shown that standard expected value arguments cannot be used to give bounds on that are exponential in , but in [8] Alon and Orlitsky proved by more sophisticated arguments that random cyclic graphs nonetheless give bounds on of order .

The triangle-free process is an important tool in studying the asymptotic lower bound on . The triangle-free process was used in studying the asymptotic lower bound for in [9, 10]. Cyclic triangle-free process (CTFP) is the cyclic analog of the triangle-free process, which is used to generate cyclic graphs of a certain order. To generate a cyclic graph of order , the process starts with an empty graph of order and iteratively adding random parameters, chosen uniformly at random, conformed to the constraint that no triangle is formed in the obtained cyclic graph, until no more parameters can be added [11].

In our previous work [11], CTFP was applied to study lower bounds on . Because of the symmetry of cyclic graphs, it is easier to compute the independence numbers of cyclic graphs than those of non-cyclic graphs with the same orders and edge density. In this paper, the previous work is extended by including an approach to generating cyclic triangle-free graphs of prime order. The experimental results demonstrate that generation of cyclic triangle-free graphs is much more efficient compared to the previous work [11].

By employing our approach, it is feasible to generate large amount of cyclic triangle-free graphs and improve some previous best known lower bounds on , including , , , and .

The remaining parts of this paper are organized as follows. In Section 2, definitions of Ramsey numbers, as well as some known results on cyclic triangle-free graphs and , are introduced. The sizes of the parameter sets of cyclic triangle-free graphs of prime order obtained by the CTFP are studied in Section 3. Furthermore, new lower bounds on for small are given in Section 4. Section 5 concludes the paper, and discusses a problem on cyclic triangle-free graphs and .

2. Preliminaries

In this section, we firstly introduce some basic concepts and notations used in this paper. Then, some basic known results on are presented. Finally, cyclic triangle-free process (CTFP) is discussed in details.

2.1. Definitions and Notations

All graphs considered in this paper are finite and undirected graphs. The complete graph of order () is denoted by . represents a triangle. For a positive integer , if every vertex in is adjacent to vertices, then is called -regular. The clique number of graph , denoted by , is the cardinality of the largest clique in . The independence number of graph , denoted by , is the cardinality of the largest independent set in . A clique of order is called a -clique, and an independent set of order is called a -independent set.

Let and be two positive integers, the Ramsey number is the smallest positive integer such that every graph of order contains either an -clique or a -independent set. In accordance with the well-known Ramsey theorem [12], it is known that is finite. An -graph is a graph that contains neither an -clique nor a -independent set. Therefore there is an -graph of order , but there is not an -graph of order .

The triangle-free process begins with , an empty graph of order , and iteratively adds edges chosen uniformly at random subject to the constraint that no triangle is formed, until no more edge can be added. The triangle-free process ends with a maximal triangle-free graph. The cyclic triangle-free process, i.e., CTFP, begins with , and generate a cyclic graph of order 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.

For any real number , we use to designate the largest integer that is smaller than or equal to . Similarly, is used to designate the smallest integer that is larger than or equal to . Given an integer , suppose , and let be a graph with the vertex set and the edge set , a graph is called a cyclic graph of order , namely . is called the parameter set of .

2.2. Some Basic Known Results on and the Triangle-Free Process

Although is simple among Ramsey numbers , it can be difficult when becomes large. The best known asymptotic lower bound on is , obtained by Bohman and Keevash [9] in 2013 and by Pontiveros et al. [10] independently and simultaneously. In 2020, the work of [10] was updated and published as [13]. The triangle-free process was used in both [9, 10]. This asymptotic lower bound on was obtained by proving the following theorem.

Theorem 1 ([9, 10]). Let be the maximal triangle-free graph of order at which the triangle-free process terminates. With high probability, has independence number at most .

The best known asymptotic lower bound on , given in [9, 10], is far from the best known upper bound , which was proved by Shearer in [14]. The exact value of is known only for positive integer . For larger small positive integer , the best known lower bound on was obtained no later than 2017, as described in the survey [15]. Most of these best known lower bounds on small were obtained by finding cyclic -graphs. The best known lower bound on for any integer cited in [15] was obtained in [16] based on cyclic triangle-free graphs.

In [17], was proved for integers and , where . The subcase , , was proved in [18] in 1989. This bound is weak in general, but it is difficult to be improved.

2.3. The Cyclic Triangle-Free Process

Most best known lower bounds on small were obtained by cyclic graphs of which the orders are composite integers. Some relevant results can be found in [16, 19, 20]. For prime orders between 120 and 260, there is only one best known lower bound obtained by cyclic graphs of prime order, which is (i.e., the lower bound on is 224) given in [16].

There is still no known interesting general lower bound on given by random cyclic graphs that is better than all linear ones. Thus, it is interesting to know if we can obtain better lower bounds on Ramsey numbers of the form by computing more triangle-free cyclic graphs generated by the CTFP. In particular, it would be interesting to study lower bounds on based on cyclic graphs of prime order. The structure of cyclic triangle-free graphs can be quite different between the prime order and the composite integer order cases. For instance, if is an odd prime, then for any cyclic graph and any , there will be a cyclic graph that is isomorphic to such that .

In our previous work, new lower bounds on some small Ramsey numbers of form , including , , and , were obtained in [11] by the CTFP, which improved the best known lower bounds in [15]. More lower bounds were obtained by the CTFP in [11], including , , , and .

It is difficult to give interesting lower bounds on small by the triangle-free process. For example, we have generated graphs of order 300 by the triangle-free process, and without difficulty, we found that all of them contain 45-independent sets. On the other hand, we have found a cyclic triangle-free graph of order 307 and independence number 40 by the CTFP. More discussions on this would be given in Section 4.

In [9, 10], it was proved that with high probability, every vertex of has degree , for the maximal triangle-free graph of order at which the triangle-free process terminates. As shown in the computing results in [11], when is not large, the graphs generated by the CTFP, compared to those generated by the triangle-free process, has more edges and smaller independence numbers with high probability. When is large, the difference between the numbers of edges in graphs generated by the CTFP, as well as those of edges in graphs generated by the triangle-free process, may become smaller.

We focus on improving the best known lower bounds on small based on cyclic triangle-free graphs of prime order. In particular, we are interesting in finding triangle-free graphs with small independence numbers. 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 CTFP in the next section.

3. The Size of Parameter Set of Cyclic Triangle-Free Graphs of Prime Order

Similar to the proof performed on the triangle-free process in [9, 10] (including Theorem 1 in Section 2.1), proving theorems on the CTFP can be difficult. Since cyclic triangle-free graphs are not well understood by now, more details on the structure of those graphs are needed.

3.1. Small Parameter Sets of Some Cyclic Triangle-Free Graphs of Prime Orders

As the order of all triangle-free graphs considered are of odd prime orders, the degree equals to the twice of the number of parameters, which is a lower bound on the independence numbers. When generating cyclic triangle-free graphs of prime orders range from 223 to 401, we have found maximal cyclic triangle-free graphs with different parameter sets. Data on sizes of parameter sets are useful in computation experiment design. The sizes of some maximal cyclic triangle-free graphs with small parameter sets are listed in Table 1. With only one exception (i.e., 151), the results on prime orders ranging from 127 to 211 are the same to those in [11].

3.2. Computing Time Through an Example

The Maximum Independence Number Problem is NP-hard even if restricted to cyclic graphs [21]. It can be difficult to compute the independence number of a large cyclic graph of which the density of edges is low. For instance, it is difficult to compute the independence number of a random cyclic triangle-free graph when the order is larger than 500.

Suppose that among all maximal cyclic triangle-free graphs of order , the proportion of graphs with independence number smaller than is . Suppose that is not too small such that holds. If we generate maximal cyclic triangle-free graphs randomly, then the probability that at least one graph among them has independence number smaller than is . When and approaches infinite, the probability tends towards 1. When and positive is very small, the probability is , where is the base of natural logarithm. To find a cyclic triangle-free graph of order and independence number smaller than with large probability, sufficient amount of cyclic triangle-free graphs should be generated. Therefore, to enable handling with large amount of graphs, the computation of graphs should be fast enough.

For any integer between 121 and 401, generating a cyclic graph of order by the CTFP costs about one second in average. However, generating a cyclic triangle-free graph with small parameter set, can spend much more time in average. When the order becomes larger, generating a cyclic graph by the CTFP can take longer time in average. On the other hand, finding a -independent set in a given cyclic triangle-free graph can be much easier when is smaller.

We discuss the computing time through an example below. By using CTFP, it is easy to find a cyclic triangle-free graph of order 313 such that . However, among 15000 cyclic graphs generated, only one cyclic graph of order 313 with is found. All of the cyclic graphs contain 20 or fewer parameters. For the triangle-free graphs of order 313, whose number of parameters is at most 20, the computation time is about one minute in average. On the other hand, almost for any graph among them, a 42-independent set can be found within one second, and we know that they can not be used to prove .

Generating random cyclic triangle-free graphs by the CTFP or similar methods quickly is important in computing new lower bounds on small . Our experiment result shows that this is not difficult when the order is an odd prime.

3.3. A Method on Generating Random Cyclic Triangle-Free Graphs of Prime Order

Suppose that is an odd prime and . Let () be a cyclic triangle-free graph, and and , is a triangle-free graph.

Based on the data on graphs generated by the CTFP, we found that among cyclic graphs of order generated that have parameters, the number of parameters in equals to or with high probability. This can be proved via Theorem 2.

Theorem 2. Suppose that is a prime and , and let be a cyclic triangle-free graph, where . Let , where and , there is a parameter set such that is isomorphic to , and .

Proof. Let and denote such that , i.e., ( is an integer). Let , and for any , and for any . For any , is isomorphic to . For any , if runs over , then runs over as well. Therefore, for any , there are sets among that contain . According to the Drawer principle, there is such that . Let be , we can conclude that .
We can also prove that in Theorem 2, with similar proof. By the method based on Theorem 2, however, we can generate cyclic triangle-free graphs more similar to those generated by the CTFP at the distribution of parameters.
The CTFP generates and in Theorem 2 with the same probability for any . Based on a result given by Muzychuk on isomorphic cyclic graphs [22], all cyclic graphs isomorphic to can be obtained in the same manner when the order is an odd prime.
Based on Theorem 2, a novel approach similar to the CTFP can be devised, which can generate cyclic graphs of prime order more effectively. We choose integers in randomly, and generate a cyclic graph of order by iteratively adding parameters. Firstly, we add parameters chosen uniformly in at random and subject to the constraint that no triangle is formed in the cyclic graph obtained, until no more parameter in can be added. Then, we iteratively add parameters that are chosen uniformly in at random and conformed to the constraint that no triangle is formed in the cyclic graph obtained, until there is no such parameter that is different from those integers chosen in earlier.
This method is similar to the CTFP. Together with other improvement, the new method can generate cyclic triangle-free graphs more quickly than the CTFP. Among cyclic graphs of order that have parameters generated using this method, in many cases, the number of parameters in equals to or .
If we generate 100 cyclic triangle-free graphs of order 313 by this method, of which the number of parameters is at most 20, then the computation spends about 28 seconds in average. This allows to generate more graphs. We have generated more than graphs of order 313 whose the number of parameters is at most 20, and found a triangle-free graph of order 313 and independence number 41. We have also generated many cyclic triangle-free graphs of order 317, of which the number of parameters is at most 20, and found a triangle-free graph of order 317 and independence number 41.
From computing results obtained by the new method, we can see that, in studying lower bounds on , the new method can achieve results similar to that derived by the CTFP more efficiently.

4. New Lower Bounds on Small

Take the Ramsey number as an example, the lower bound was obtained in [11]. If we can improve it into based on a cyclic triangle-free graph of order 257, then has at most 18 parameters. It is easy to find a large independent set in some cases, in which the neighborhood of a vertex is a 36-independent set contained in a larger independent set. Hence in these cases, and graphs obtained can not be used in improving .

This method is powerful in computing graphs when the orders of cyclic graphs are large. Although useful, this method is less important in improving the efficiency as the method on generating random cyclic triangle-free graphs, as discussed in the last section.

We have conducted much computation to improve the best known lower bound on , based on graphs of prime order , generated by the CTFP and the new method described in the last section. For each prime that ranges from 127 to 401, we have generated more than 10000 cyclic graphs of order , of which the amount of parameters is small enough to enable improving the best known lower bound on .

In [11], a question was proposed for cases between 200 and 230, i.e., whether additional computation using the CTFP can improve the best known lower bounds on . We have carried out much computation for any prime among , and have found a cyclic triangle-free graph of order 229 and independence number 33. Therefore , which improves the previous best known lower bound on by 1.

Table 2 presents a list of the smallest independence numbers of generated graphs of order . In most cases, the results were obtained based on graphs generated by the CTFP, and in some cases were obtained based on graphs generated by the new method presented in the last section. Some results of small prime orders were known before, including , , , , , and . Although improving lower bounds on in Table 2 can be difficult, by the new method on generating random cyclic triangle-free graphs, obtaining the same lower bounds on is easier.

We have also generated more than graphs of order 197 of which the number of parameters is at most 14 by the new method, and the independence numbers are all larger than 29. Note that is the best known lower bound on given in [16]. We have not found a cyclic triangle-free graph of order 389 and independence number 47, among the generated graphs whose the number of parameters is at most 23, generated using the new method.

Some new lower bounds on , obtained based on the results in Table 2, are listed in Theorem 3.

Theorem 3. , , , , , , , , , , , , , , and can be obtained by cyclic triangle-free graphs.

Compared to the results given in [11], except the first one (i.e., ), all the best known lower bound are improved by 5 or more. We list the parameter sets and independence numbers of some graphs obtained by the CTFP or the similar method in Table 3, based on which the result in Theorem 3 is obtained.

For the cases in Table 3 where the order , the lower bound is better when the independence number is even. The reason is given below: if the expected independence number is even, we can generate cyclic triangle-free graphs of which the number of parameters is no larger than quickly, which allows to deal with many graphs; if more random cyclic triangle-free graphs are generated, we can improve some best known lower bounds of form in which cases the expected independence number is odd.

5. Conclusion and Discussions

In this paper, we have improved the best known lower bound for based on some cyclic graphs of prime order obtained by the CTFP or a similar method. For that is not small, the works on the lower bound for . based on cyclic triangle-free graphs earlier than [11] were not efficient in finding good parameter sets. The CTFP can be used as a good tool in studying the lower bound on for large .

We propose a problem on cyclic triangle-free graphs of prime orders.

Problem 1. Suppose that is an integer and . Let be the minimum among the independence numbers of all cyclic triangle-free graphs of order . Is there an integer such that for any pair primes and , when ?

We propose this problem based on the data in Table 2. For instance, we know that and , while whether is unknown. Note that in this paper we have obtained lower bounds on by computing small upper bounds for .

For a positive integer , there is no cyclic -graphs of order that can be used to improve the best known lower bound on given in [15] (see [23]). It is likely that for any integer between 121 and 200, there is not a cyclic triangle-free graph that can be used to improve the best known lower bound on .

Data Availability

The data used to support the findings of this study are included within the article.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

Research is supported in part by the National Natural Science Foundation of China (11361008).

References

R. L. Graham, B. L. Rothschild, and J. H. Spencer, Ramsey Theory, John Wiley & Sons, Hoboken, NJ, USA, 1990.
J. Han, M. Jenssen, Y. Kohayakawa, G. O. Mota, and B. Roberts, “The multicolour size-ramsey number of powers of paths,” Journal of Combinatorial Theory, Series B, vol. 145, pp. 359–375, 2020.
View at: Publisher Site | Google Scholar
L. Lu and Z. Wang, “On the cover ramsey number of berge hypergraphs,” Discrete Mathematics, vol. 343, no. 9, p. 111972, 2020.
View at: Publisher Site | Google Scholar
F. Molnár, S. R. Kharel, X. S. Hu, and Z. Toroczkai, “Accelerating a continuous-time analog SAT solver using GPUs,” Computer Physics Communications, vol. 256, p. 107469, 2020.
View at: Publisher Site | Google Scholar
A. Raigorodskii and M. Koshelev, “New bounds on clique-chromatic numbers of johnson graphs,” Discrete Applied Mathematics, vol. 283, pp. 724–729, 2020.
View at: Publisher Site | Google Scholar
X. Xu, M. Liang, and H. Luo, Some Unsolved Problems and Results in Ramsey Theory, Walter de Gruyter GmbH, Berlin, Germany, 2018.
N. J. Calkin, P. Erdos, and C. A. Tovey, “New ramsey bounds from cyclic graphs of prime order,” SIAM Journal on Discrete Mathematics, vol. 10, no. 3, pp. 381–387, 1997.
View at: Publisher Site | Google Scholar
N. Alon and A. Orlitsky, “Repeated communication and ramsey graphs,” IEEE Transactions on Information Theory, vol. 41, no. 5, pp. 1276–1289, 1995.
View at: Publisher Site | Google Scholar
T. Bohman and P. Keevash, “Dynamic concentration of the triangle-free process,” The Seventh European Conference on Combinatorics, Graph Theory and Applications, Springer, Berlin, Germany, 2013.
View at: Publisher Site | Google Scholar
G. F. Pontiveros, S. Griffiths, and R. Morris, “The triangle-free process and R(3, k),” 2013, https://arxiv.org/abs/1302.6279.
View at: Google Scholar
Y. Jiang, M. Liang, Y. Teng, and X. Xu, “The cyclic triangle-free process,” Symmetry, vol. 11, no. 8, p. 955, 2019.
View at: Publisher Site | Google Scholar
F. P. Ramsey, “On a problem of formal logic,” Proceedings of the London Mathematical Society, vol. s2–30, no. 1, pp. 264–286, 1930.
View at: Publisher Site | Google Scholar
G. F. Pontiveros, S. Griffiths, and R. Morris, “The triangle-free process and the Ramsey number R(3, k),” American Mathematical Society, vol. 263, no. 1274, pp. 1–138, 2020.
View at: Publisher Site | Google Scholar
J. B. Shearer, “A note on the independence number of triangle-free graphs,” Discrete Mathematics, vol. 46, no. 1, pp. 83–87, 1983.
View at: Publisher Site | Google Scholar
S. Radziszowski, “Small ramsey numbers,” The Electronic Journal of Combinatorics, vol. 15, pp. 1–104, 2017, https://www.combinatorics.org/files/Surveys/ds1/ds1v15-2017.pdf.
View at: Publisher Site | Google Scholar
L. Mingbo and L. Yusheng, “Ramsey numbers and triangle-free cayley graphs,” Journal of Tongji University (Natural Science), vol. 11, p. 22, 2015.
View at: Google Scholar
X. Xu, Z. Xie, and S. P. Radziszowski, “A constructive approach for the lower bounds on the ramsey numbers R(s, t),” Journal of Graph Theory, vol. 47, no. 3, pp. 231–239, 2004.
View at: Publisher Site | Google Scholar
S. A. Burr, P. Erdos, R. J. Faudree, and R. Schelp, “On the difference between consecutive ramsey numbers,” Utilitas Mathematica, vol. 35, pp. 115–118, 1989.
View at: Google Scholar
K. Wu, W. Su, H. Luo, and X. Xu, “New lower bounds for seven classical ramsey numbers R(3, q),” Applied Mathematics Letters, vol. 22, no. 3, pp. 365–368, 2009.
View at: Publisher Site | Google Scholar
K. Wu, W. Su, H. Luo, and X. Xu, “A generalization of generalized paley graphs and new lower bounds for R(3, q),” The Electronic Journal of Combinatorics, vol. 17, no. 1, p. N25, 2010.
View at: Publisher Site | Google Scholar
B. Codenotti, I. Gerace, and S. Vigna, “Hardness results and spectral techniques for combinatorial problems on circulant graphs,” Linear Algebra and Its Applications, vol. 285, no. 1–3, pp. 123–142, 1998.
View at: Publisher Site | Google Scholar
M. Muzychuk, “Ádám’s conjecture is true in the square-free case,” Journal of Combinatorial Theory, Series A, vol. 72, no. 1, pp. 118–134, 1995.
View at: Publisher Site | Google Scholar
F. Deng, Z. Shao, and X. Xu, “An algorithm for finding optimal lower bounds on Ramsey numbers based on cyclic graphs,” Journal of Computational and Theoretical Nanoscience, vol. 9, no. 10, pp. 1603–1605, 2012.
View at: Publisher Site | Google Scholar

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Copyright © 2021 Yu Jiang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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