1 Introduction

ANNUAIRE DE L’UNIVERSITÉ DE SOFIA “St. Kl. OHRIDSKI”

[3pt] FACULTÉ DE MATHÉMATIQUES ET INFORMATIQUE

SMALL MINIMAL (3, 3)-RAMSEY GRAPHS

ALEKSANDAR BIKOV

We say that is a -Ramsey graph if every -coloring of the edges of forces a monochromatic triangle. The -Ramsey graph is minimal if does not contain a proper -Ramsey subgraph. In this work we find all minimal -Ramsey graphs with up to 13 vertices with the help of a computer, and we obtain some new results for these graphs. We also obtain new upper bounds on the independence number and new lower bounds on the minimum degree of arbitrary -Ramsey graphs.

Keywords. Ramsey graph, clique number, independence number, chromatic number
2000 Math. Subject Classification. 05C55

1 Introduction

In this work only finite, non-oriented graphs without loops and multiple edges are considered. The following notations are used:

- the vertex set of ;

- the edge set of ;

- the complement of ;

- the clique number of ;

- the independence number of ;

- the chromatic number of ;

- the set of all vertices of G adjacent to ;

- the degree of the vertex , i.e. ;

- subgraph of induced by ;

- subgraph of obtained from by deleting the vertex and all edges incident to ;

- subgraph of obtained from by deleting the edge ;

- the maximum degree of ;

- the minimum degree of ;

- complete graph on vertices;

- simple cycle on vertices;

- graph for which and , where , i.e. is obtained by connecting every vertex of to every vertex of .

All undefined terms can be found in [13].

Each partition

(1.1)

is called an -coloring of the edges of . We say that is a monochromatic subgraph of color in the -coloring (1.1), if .

Let and be positive integers, and . The expression means that for every -coloring of there exists a -clique of the first color or a -clique of the second color. If , we say that is a -Ramsey graph. Similarly, the expression is defined for the -colorings of .

The smallest possible integer for which is called a Ramsey number and is denoted by . The Ramsey numbers are defined similarly.

The existence of Ramsey numbers was proved by Ramsey in [32]. Only a few exact values of Ramsey numbers are known (see [30]). In this work we will use the equality . This equality means that and . It is clear, that if , then . In [6] Erdös and Hajnal posed the following problem:

Is there a graph with ?

Figure 1: The complement of the Pósa graph from [12]

The first example of a graph which gives a positive answer to this question was given by Pósa. The complement of this graph is presented on Figure 1. Pósa did not publish this result himself, but the graph was included in [12]. Later, Graham [11] constructed the smallest possible example of such a graph, namely . It is easy to see that the Pósa graph contains (it is the subgraph induced by the black vertices on Figure 1).

There exist -Ramsey graphs which do not contain . These graphs have at least 15 vertices [29]. The first 15-vertex -Ramsey graph which does not contain was constructed by Nenov [25]. This graph is obtained from the graph presented on Figure 2 by adding a new vertex which is adjacent to all vertices of .

Figure 2: The Nenov graph from [25]

Folkman constructed a graph with [7]. The minimum number of vertices of such graphs is not known. To date, we know only that this minimum is between 19 and 786, [31] and [18].

Obviously, if is a -Ramsey graph, then its every supergraph is also a -Ramsey graph.

Definition 1.1.

We say that is a minimal -Ramsey graph if and for each proper subgraph of .

It is easy to see that is a minimal -Ramsey graph and there are no minimal -Ramsey graphs with 7 vertices. The only such 8-vertex graph is the Graham graph , and there is only one such 9-vertex graph, Nenov [22] (see Figure 4).

Figure 3: 9-vertex minimal -Ramsey graph
Figure 4: 10-vertex minimal -Ramsey graph

For each pair of positive integers , there exist infinitely many minimal -Ramsey graphs [2], [8]. The simplest infinite sequence of minimal -Ramsey graphs are the graphs . This sequence contains the already mentioned graphs and . It was obtained by Nenov and Khadzhiivanov in [27]. Later, this sequence was reobtained in [3], [9], [35].

Three 10-vertex minimal -Ramsey graphs are known. One of them is from the sequence . The other two were obtained by Nenov in [24] (the second graph is presented on Figure 4 and the third is a subgraph of ).

The main goal of this work is to find new minimal -Ramsey graphs. To achieve this, we develop computer algorithms which are presented in Section 3. Using Algorithm 3.1, in Section 4 we find all minimal -Ramsey graphs with up to 12 vertices. In the next Section 5 we find all 13-vertex minimal -Ramsey graphs using Algorithm 3.11. From the graphs found in Section 4 and Section 5 we obtain interesting corollaries, which are presented in Section 6. In Section 7 and Section 8, with the help of Algorithm 3.8 we obtain, accordingly, new upper bounds on the independence number and new lower bounds on the minimum degree of minimal -Ramsey graphs with an arbitrary number of vertices.

Similar computer aided research is made in [17], [29], [4], [5], [31], [36], [18] and [34]. We shall note that the algorithms from [29] were very useful to us.

This work is an extended version of my Master’s thesis under the supervision of prof. Nedyalko Nenov. The most essential new element is Algorithm 3.8, which is obtained jointly with prof. Nenov.

2 Auxiliary results

We will need the following results:

Theorem 2.1.

[2][8] Let be a minimal -Ramsey graph. Then, . In particular, when , we have .

Definition 2.2.

We say that is a Sperner graph if for some pair of vertices .

Proposition 2.3.

If is a minimal -Ramsey graph, then is not a Sperner graph.

Proof.

Suppose the opposite is true, and let be such that . We color the edges of with two colors in such a way that there is no monochromatic -clique of the first color and no monochromatic -clique of the second color. After that, for each vertex we color the edge with the same color as the edge . We obtain a 2-coloring of the edges of with no monochromatic -cliques of the first color and no monochromatic -cliques of the second color. ∎

Theorem 2.4.

[29] Let be a -Ramsey graph and . If , then .

Figure 5: 14-vertex minimal -Ramsey graph with a single 5 clique

According to Theorem 2.4, every -Ramsey graph with no more than 14 vertices contains a 5-clique. There exist 14-vertex -Ramsey graphs containing only a single 5-clique, an example of such a graph is presented on Figure 5. The graph on Figure 5 is obtained with the help of the only 15-vertex bicritical -Ramsey graph with clique number 4 from [29]. First, by removing a vertex from the bicritical graph, we obtain 14-vertex graphs without 5 cliques. After that, by adding edges to the obtained graphs, we find a 14-vertex -Ramsey graph with a single 5-clique whose subgraph is the minimal -Ramsey graph on Figure 5. Let us note that in [29] they obtain all 15-vertex -Ramsey graphs with clique number 4, and with the help of these graphs, one can find more examples of 14-vertex -Ramsey graphs.

Theorem 2.5.

[19] Let be a graph and . Then, . In particular, if , then .

Corollary 2.6.

Let , let be independent vertices of and . Then, .

Theorem 2.7.

Let be a minimal -Ramsey graph. Then, for each vertex we have .

Proof.

Suppose the opposite is true, and let be an independent set in such that . Let . Consider a 2-coloring of the edges of in which there are no monochromatic triangles. We color the edges and with the same color in such a way that there is no monochromatic triangle (if and are adjacent, we chose the color of and to be different from the color of , and if and are not adjacent, then we chose an arbitrary color for and ). We color the remaining edges incident to with the other color, which is different from the color of and . Since is and independent set, we obtain a 2-coloring of the edges of without monochromatic triangles, which is a contradiction. ∎

Corollary 2.8.

Let be a minimal -Ramsey graph and for some vertex . Then, .

3 Algorithms

In this section, the computer algorithms used in this work are presented.

The first algorithm is appropriate for finding all minimal -Ramsey graphs with a small number of vertices.

Algorithm 3.1.

Finding all minimal -Ramsey graphs with vertices, where is fixed and .

1. Generate all n-vertex non-isomorphic graphs with minimum degree at least 4, and denote the obtained set by .

2. Remove from all Sperner graphs.

3. Remove from all graphs with clique number not equal to 5.

4. Remove from all graphs with chromatic number less than 6.

5. Remove from all graphs which are not -Ramsey graphs.

6. Remove from all graphs which are not minimal -Ramsey graphs.

Theorem 3.2.

Fix . Then, after executing Algorithm 3.1, consists of all -vertex minimal -Ramsey graphs.

Proof.

Step 6 guaranties that contains only minimal -Ramsey graphs with vertices. Let be an arbitrary -vertex minimal -Ramsey graph. We will prove that . By Theorem 2.1, , and by Theorem 2.3, is not a Sperner graph. Since , by Theorem 2.4 we have . By Theorem 2.5, . Therefore, after step 4, . ∎

In section 4 of this work we use Algorithm 3.1 to obtain all -Ramsey graphs with up to 12 vertices. Algorithm 3.1 is not appropriate in the cases , because the number of graphs generated in step 1 is too big. To find the 13-vertex minimal -Ramsey graphs, we will use Algorithm 3.11, which is defined below.

In order to present the next algorithms we will need the following definitions and auxiliary propositions:

We say that a 2-coloring of the edges of a graph is -free if it has no monochromatic triangles.

Definition 3.3.

Let be a graph and . Let be a graph which is obtained by adding a new vertex to such that . We say that is a marked vertex set in if there exists a -free -coloring of the edges of which cannot be extended to a -free -coloring of the edges of .

It is clear that if , then there are no marked vertex sets in . The following proposition is true:

Proposition 3.4.

Let be a minimal -Ramsey graph, let be independent vertices of and . Then, , are marked vertex sets in .

Proof.

Suppose the opposite is true, i.e. is not a marked vertex set in for some . Since is a minimal -Ramsey graph, there exists a -free -coloring of the edges of , which induces a -free -coloring of the edges of . By supposition, we can extend this -coloring to a -free -coloring of the edges of the graph . Thus, we obtain a -free -coloring of the edges of , which is a contradiction. ∎

Definition 3.5.

Let be a family of marked vertex sets in the graph . Let be a graph which is obtained by adding a new vertex to such that . We say that is a complete family of marked vertex sets in , if for each -free -coloring of the edges of there exists such that this -coloring can not be extended to a -free -coloring of the edges of .

Proposition 3.6.

Let be independent vertices of the graph and . If is a complete family of marked vertex sets in , then .

Proof.

Consider a 2-coloring of the edges of which induces a 2-coloring with no monochromatic triangles in . According to Definition 3.5, this 2-coloring of the edges of can not be extended in without forming a monochromatic triangle. ∎

It is easy to prove the following strengthening of Proposition 3.4:

Proposition 3.7.

Let be a minimal -Ramsey graph, let be independent vertices of and . Then, is a complete family of marked vertex sets in . What is more, this family is a minimal complete family, in the sense that it does not contain a proper complete subfamily.

Let be a minimal -Ramsey graph and . Let be an independent set in such that . Then, , and therefore the graph is obtained by adding and independent set of vertices to the -vertex graph . From Proposition 2.3 it is easy to see that for a fixed there are a finite number of minimal -Ramsey graphs for which . We define an algorithm for finding all minimal -Ramsey graphs for which , where is fixed (but is not fixed).

Algorithm 3.8.

(A. Bikov and N. Nenov) Finding all minimal -Ramsey graphs for which and , where and are fixed positive integers.

1. Denote by the set of all -vertex graphs for which and . The obtained minimal -Ramsey graphs will be output in the set , let .

2. For each graph :

2.1. Find all subsets of which have the properties:

(a) , i.e. is a -free subset.

(b) .

(c) is a marked vertex set in (see Definition 3.3).

Denote by the family of subsets of which have the properties (a), (b) and (c). Enumerate the elements of : .

2.2. Find all minimal complete subfamilies of (see Definition 3.5). For each such found subfamily construct the graph by adding new independent vertices to such that . Add to . If there are no complete subfamilies of , then no supergraphs of are added to .

3. Remove isomorph copies of graphs from .

4. Remove from all non-minimal -Ramsey graphs.

Remark 3.9.

It is clear, that if is a minimal -Ramsey graph and , then . Obviously there are no -Ramsey graphs with clique number less than 3. Therefore, we shall use Algorithm 3.8 only for .

Theorem 3.10.

After executing Algorithm 3.8, the set coincides with the set of all minimal -Ramsey graphs for which and .

Proof.

From step 2.2 it becomes clear that every graph which is added to is obtained by adding independent vertices to a graph . Therefore, . From and , it follows . According to Proposition 3.6, after step 2.2 contains only -Ramsey graphs, and after step 4 contains only minimal -Ramsey graphs.

In order to prove that contains all minimal -Ramsey graphs which fulfill the conditions, consider an arbitrary minimal -Ramsey graph for which and . We will prove that .

Denote . Let be independent vertices of and . By 2.6, . Therefore, after executing step 1, .

From it follows . By Proposition 2.3, is not a Sperner graph, and therefore . According to Proposition 3.4, are marked vertex sets in . Therefore, after executing step 2.1, .

From Proposition 3.7 it becomes clear that is a minimal complete subfamily of . Therefore, in step 2.2 the graph is added to .

Thus, the theorem is proved. ∎

In order to find the 13-vertex minimal -Ramsey graphs we will use the following modification of Algorithm 3.8 in which is fixed:

Algorithm 3.11.

Modification of Algorithm 3.8 for finding all -vertex minimal -Ramsey graphs for which and , where , and are fixed positive integers.

In step 2.2 of Algorithm 3.8 add the condition to consider only minimal complete subfamilies of in which .

4 Minimal -Ramsey graphs with up to 12 vertices

We execute Algorithm 3.1 for , and we find all minimal -Ramsey graphs with up to 12 vertices except . In this way, we obtain the known results: there is no minimal -Ramsey graph with 7 vertices, the Graham graph is the only such 8-vertex graph, and there exists only one such 9-vertex graph, the Nenov graph from [22] (see Figure 4). We also obtain the following new results:

Theorem 4.1.

There are exactly 6 minimal 10-vertex -Ramsey graphs. These graphs are given on Figure 14, and some of their properties are listed in Table 5.

Theorem 4.2.

There are exactly 73 minimal 11-vertex -Ramsey graphs. Some of their properties are listed in Table 5. Examples of 11-vertex minimal -Ramsey graphs are given on Figure 15 and Figure 16.

Theorem 4.3.

There are exactly 3041 minimal 12-vertex -Ramsey graphs. Some of their properties are listed in Table 5. Examples of 12-vertex minimal -Ramsey graphs are given on Figure 18 and Figure 18.

We will use the following enumeration for the obtained minimal -Ramsey graphs:
- , …, are the 10-vertex graphs;
- , …, are the 11-vertex graphs;
- , …, are the 12-vertex graphs;

The indexes correspond to the order of the graphs’ canonical labels defined in nauty [20].

Detailed data for the number of graphs obtained at each step of the execution of Algorithm 3.1 is given in Table 1.

Step of
Algorithm 3.1
1 424 15 471 1 249 973 187 095 840 48 211 096 031
2 59 2 365 206 288 33 128 053 9 148 907 379
3 9 380 41 296 8 093 890 2 763 460 021
4 1 7 356 78 738 44 904 195
5 1 3 126 23 429 11 670 079
6 1 1 6 73 3041
Table 1: Steps in finding all minimal -Ramsey graphs with up to 12 vertices
30 1 4 1 9 6 2 3 6 6 4 2
31 1 5 4 3 3 8 2
32 2 6 1 16 1
33 1 84 1
34 1
Table 2: Some properties of the 10-vertex minimal -Ramsey graphs
35 6 4 5 8 1 2 4 6 73 1 20
36 13 5 58 10 72 3 66 2 29
37 23 6 10 4 3 4 14
38 25 6 1
39 5 8 4
41 1 12 1
16 3
24 1
Table 3: Some properties of the 11-vertex minimal -Ramsey graphs
38 5 4 129 8 43 2 124 6 3 041 1 1 792
39 27 5 2 178 9 1 196 3 2 431 2 851
40 144 6 611 11 1 802 4 485 4 286
41 418 7 123 5 1 6 1
42 1 014 8 67
43 459 12 16
44 224 16 18
45 351 24 6
46 299 32 1
47 84 36 1
48 16 96 1
108 1
Table 4: Some properties of the 12-vertex minimal -Ramsey graphs
41 4 4 13 725 8 16 2 13 6 306 622 1 251 976
42 44 5 191 504 9 61 678 3 218 802 7 13 2 46 487
43 220 6 85 932 10 175 108 4 86 721 3 10
44 1 475 7 15 391 12 69 833 5 1 097 4 6 851
45 7 838 8 83 6 2 6 83
46 28 805 8 916
47 33 810 12 129
48 26 262 16 106
49 39 718 24 44
50 62 390 32 12
51 59 291 36 3
52 34 132 40 1
53 10 878 48 11
54 1 680 72 3
55 86 96 2
56 2 144 1
Table 5: Some properties of the 13-vertex minimal -Ramsey graphs

5 Minimal -Ramsey graphs with 13 vertices

The method with which we find all 13-vertex minimal -Ramsey graphs consists of two parts:

1. First, we find the 13-vertex minimal -Ramsey graphs with independence number 2. We use [30], and that all graphs for which and are known [21]. Among them, the 13-vertex graphs are 275 086. By computer check, we find that exactly 13 of these graphs are minimal -Ramsey graphs.

2. It remains to find the 13-vertex minimal -Ramsey graphs with independence number at least 3. To do this, we execute Algorithm 3.11(). First, in step 1 of Algorithm 3.11 we find all 1 923 103 graphs with 10 vertices for which and . After that, in step 2 of Algorithm 3.11 we add 3 independent vertices to the obtained 10-vertex graphs, and thus, we obtain all 306 622 minimal -Ramsey graphs with 13-vertices and independence number at least 3.

Finally, we obtain

Theorem 5.1.

There are exactly 306 635 minimal 13-vertex -Ramsey graphs. Some of their properties are listed in 5. Examples of 13-vertex minimal -Ramsey graphs are given on Figure 6, Figure 20 and Figure 21.

We enumerate the obtained 13-vertex -Ramsey graphs: , …, .

As noted, all graphs for which and are known and from it follows that these graphs have at most 17 vertices. By computer check we find that there are no minimal -Ramsey graphs with independence number 2 and more than 13 vertices. Thus, we prove

Theorem 5.2.

Let be a minimal -Ramsey graph and . Then, . There are exactly 145 minimal -Ramsey graphs for which :

- 8-vertex: 1 ();

- 9-vertex: 1 (see Figure 4);

- 10-vertex: 3 (, , , see Figure 14);

- 11-vertex: 4 (, , , , see Figure 16);

- 12-vertex: 124;

- 13-vertex: 13 (see Figure 21);

By executing Algorithm 3.11(), we find all minimal -Ramsey graphs with 10, 11 and 12 vertices and independence number greater than 2. In this way, with the help of Theorem 5.2, we obtain a new proof of Theorem 4.1, Theorem 4.2 and Theorem 4.3.

6 Corollaries from the obtained results

6.1 Minimum and maximum degree

Figure 6: 8-regular 13-vertex minimal -Ramsey graph

By Theorem 2.1, if is a minimal -Ramsey graph, then . Via very elegant constructions, in [2] and [8] it is proved that the bound from Theorem 2.1 is exact. However, these constructions are not very economical in the case . For example, the minimal -Ramsey graph from [8] with is not presented explicitly, but it is proved that it is a subgraph of a graph with 17577 vertices. From the next theorem we see that the smallest minimal -Ramsey graph with has 10 vertices:

Theorem 6.1.

Let be a minimal -Ramsey graph and . Then, . There is only one 10-vertex minimal -Ramsey graph with , namely (see Figure 14). What is more, has only a single vertex of degree 4. For all other 10-vertex minimal -Ramsey graphs , .

Let be a -Ramsey graph. By Theorem 2.5, and from the inequality (see [13]) we obtain . From the Brooks’ Theorem (see [13]) it follows that if , then . The following related question arises naturally:

Are there minimal -Ramsey graphs which are 6-regular?
(i.e. )

From the obtained minimal -Ramsey graphs we see that the following theorem is true:

Theorem 6.2.

Let be a regular minimal -Ramsey graph and . Then, . There is only one regular minimal -Ramsey with 13 vertices, and this is the graph presented on Figure 6, which is 8-regular.

In regard to the maximum degree of the minimal -Ramsey graphs we obtain the following result:

Theorem 6.3.

Let be a minimal -Ramsey graph. Then:

(a) , if .

(b) , if , or .

6.2 Chromatic number

By Theorem 2.5, if is a -Ramsey graph, then .

From the obtained minimal -Ramsey graphs we derive the following results:

Theorem 6.4.

Let be a minimal -Ramsey graph and . Then .

Theorem 6.5.

Let G be a minimal -Ramsey graph and . Then . The smallest 7-chromatic minimal -Ramsey graphs are the 13 minimal -Ramsey graph with 13 vertices and independence number 2, given on Figure 21.

Proof.

Suppose the opposite is true, i.e. . Then, according to [26], , where is the graph presented on Figure 7. The graph is a -Ramsey graph, but it is not minimal. By Theorem 6.4, there are no 7-chromatic minimal -Ramsey graphs with less than 13 vertices. The graphs on Figure 21 are 13-vertex minimal -Ramsey graphs with independence number 2, and therefore these graphs are 7-chromatic. By computer check, we find that among the 13-vertex -Ramsey graphs with independence number greater than 2 there are no 7-chromatic graphs. ∎

Figure 7: Graph

6.3 Multiplicities

Definition 6.6.

Denote by the minimum number of monochromatic triangles in all 2-colorings of . The number is called a -multiplicity of the graph .

In [10] the -multiplicities of all complete graphs are computed, i.e. is computed for all positive integers . Similarly, the -multiplicity of a graph is defined [14]. The following works are dedicated to the computation of the multiplicities of some concrete graphs: [15], [16], [33], [1], [28].

With the help of a computer, we check the -multiplicities of the obtained minimal -Ramsey graphs and we derive the following results:

Theorem 6.7.

If is a minimal -Ramsey graph, and , then .

We suppose the following hypothesis is true:

Hypothesis 6.8.

If is a minimal -Ramsey graph and , then .

In support to this hypothesis we prove the following:

Proposition 6.9.

If is a minimal -Ramsey graph, and , then .

Proof.

Let and . Consider a 2-coloring of without monochromatic triangles. We will color the edges incident to with two colors in such a way that we will obtain a 2-coloring of with exactly one monochromatic triangle. To achieve this, we consider the following two cases:

Case 1: . By Corollary 2.8, . Let and suppose that is colored with the first color. Then, is also colored with the first color (otherwise, by coloring and with the second color and and with the fist color, we obtain a 2-coloring of without monochromatic triangles). Thus, and are colored in the first color. We color and with the first color and and with the second color. We obtain a 2-coloring of with exactly one monochromatic triangle .

Case 2: . Since , in there are two non-adjacent vertices and . From it follows easily that in there is an edge of the first color and an edge of the second color. Therefore, we can suppose that in there is exactly one edge of one of the colors, say the first color. We color and with the second color and the other three edges incident to with the first color. We obtain a 2-coloring of with exactly one monochromatic triangle. ∎

In the end, also in support to the hypothesis, we shall note that [27].

6.4 Automorphism groups

Denote by the automorphism group of the graph . We use the nauty programs [20] to find the number of automorphisms of the obtained minimal -Ramsey graphs with 10, 11, 12 and 13 vertices. Most of the obtained graphs have small automorphism groups (see Table 5, Table 5, Table 5 and Table 5). We list the graphs with at least 60 automorphisms:

- The graphs in the form : . , ;

- (see Figure 18);

- , , , , , (see Figure 20);

7 Upper bounds on the independence number of the minimal -Ramsey graphs

Figure 8: 14-vertex minimal-Ramsey graphwith independence number 7
Figure 9: 26-vertex minimal-Ramsey graphwith independence number 18
Figure 10: 29-vertex minimal -Ramsey graph
with clique number 4 and independence number 20

In regard to the maximal possible value of the independence number of the minimal -Ramsey graphs, the following theorem holds:

Theorem 7.1.

[23] If is a minimal -Ramsey graph, and , then . There is a finite number of graphs for which equality is reached.

From Theorem 7.1 it follows that by executing Algorithm 3.8() we obtain all minimal -Ramsey graphs for which or . As a result of the execution of this algorithm we derive the following additions to Theorem 7.1:

Theorem 7.2.

There are exactly 11 minimal -Ramsey graphs , for which :

- 9-vertex: 1 (Figure 4);

- 10-vertex: 3 (, , , see Figure 14);

- 11-vertex: 3 (, , , see Figure 15);

- 12-vertex: 1 (, see Figure 18);

- 13-vertex: 2 (, , see Figure 19) ;

- 14-vertex: 1 (see Figure 10);

Theorem 7.3.

There are exactly 8633 minimal -Ramsey graphs for which . The largest of these graphs has 26 vertices, and it is given on Figure 10. There is only one minimal -Ramsey graph for which and , and it is the 15-vertex graph from [25] (see Figure 2).

Corollary 7.4.

Let be a minimal -Ramsey graph and . Then, .

According to Theorem 7.3, if is a minimal -Ramsey graph, , and , then . From Theorem 2.4 it follows that by executing Algorithm 3.8() we obtain all minimal -Ramsey graphs for which and , and the graph . As a result of the execution of this algorithm we derive:

Theorem 7.5.

There are exactly 8903 minimal -Ramsey graphs for which and . The largest of these graphs has 29 vertices, and it is given на Figure 10.

Corollary 7.6.

Let be a minimal -Ramsey graph such that and . Then, .

8 Lower bounds on the minimum degree of the minimal -Ramsey graphs

According to Proposition 3.4, if is a minimal -Ramsey graph, then for each vertex of , is a marked vertex set in , and therefore is a marked vertex set in .

Figure 11: -free 2-coloring of the edges of

It is easy to see that if and , or and , then is not a marked vertex set in . A -free 2-coloring of which cannot be extended to a -free 2-coloring of is shown on Figure 11. Therefore, the only 4-vertex graph such that is a marked vertex set in is .

Figure 12: The graphs , ,

With the help of a computer, we obtain that there are exactly 3 graphs with 5 vertices such that and is a marked vertex set in . Namely, they are the graphs , and presented on Figure 12. We shall note that . From these results we derive

Theorem 8.1.

Let be a minimal -Ramsey graph and . Then, . If and , then for some (see Figure 12).

The bound from Theorem 8.1 is exact. For example, the graph from [25] (see Figure 2) has 7 vertices such that and .

Figure 13: The graphs

Also with the help of a computer, we obtain that the smallest graphs such that and is a marked vertex set in have 8 vertices, and there are exactly 7 such graphs. Namely, they are the graphs presented on Figure 13. Among them, the minimal graphs are , and , and the remaining 4 graphs are their supergraphs. Thus, we derive the following

Theorem 8.2.

Let be a minimal -Ramsey graph and . Then, . If and , then for some (see Figure 13).

ACKNOWLEDGEMENTS

I would like to thank prof. Nedyalko Nenov, who read the manuscript and made some suggestions which led to the improvement of this work.

APPENDICES

A. GRAPHS

Figure 14: 10-vertex minimal -Ramsey graphs
Figure 15: 11-vertex minimal -Ramsey graphs
with independence number 4
Figure 16: 11-vertex minimal -Ramsey graphs
with independence number 2
Figure 17: 12-vertex minimal -Ramsey graphwith independence number 5
Figure 18: 12-vertex minimal -Ramsey graphwith 96 automorphisms
Figure 19: 13-vertex minimal -Ramsey graphs
with independence number 6
Figure 20: 13-vertex minimal -Ramsey graphs
with a large number of automorphisms
Figure 21: 13-vertex minimal -Ramsey graphs
with independence number 2

REFERENCES


Faculty of Mathematics and Informatics

”St. Kl. Ohridski” University of Sofia

5  J. Bourchier blvd., BG-1164 Sofia

BULGARIA

e-mail: asbikov@fmi.uni-sofia.bg

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