Computation of the Ramsey Numbers R( C 4 , K 9 ) and R( C 4 , K 10 )
CComputation of the Ramsey Numbers R ( C , K ) and R ( C , K ) Ivan Livinsky
Department of MathematicsUniversity of TorontoToronto, ON M5S 2E4 [email protected]
Alexander Lange, Stanis(cid:32)law Radziszowski
Department of Computer ScienceRochester Institute of TechnologyRochester, NY 14623 { arl9577,spr } @cs.rit.edu Abstract
The Ramsey number R ( C , K m ) is the smallest n such that any graph on n vertices contains a cycle of length four or an independent set of order m . Withthe help of computer algorithms we obtain the exact values of the Ramseynumbers R ( C , K ) = 30 and R ( C , K ) = 36. New bounds for the next twoopen cases are also presented. Let G and H be simple graphs. An n -vertex graph F is a ( G, H ; n )-graph if it containsno subgraph isomorphic to G and F contains no subgraph isomorphic to H . Define R ( G, H ; n ) to be the set of all such graphs. The Ramsey number R ( G, H ) is thesmallest n such that for every two-coloring of the edges of K n , a monochromatic copyof G or H exists in the first or second color, respectively. Clearly, if a ( G, H ; n )-graphexists, then R ( G, H ) > n . It is known that Ramsey numbers exist [20] for all G and H . The values and bounds for various types of such numbers are collected andregularly updated by the third author [18].The cycle-complete Ramsey numbers R ( C n , K m ) have received much attention,both theoretically and computationally. For fixed n = 3, the problem becomesthat of R (3 , k ), which has been widely studied (see for example [24]), including ex-act determination of its asymptotics [14]. Since 1976, it has been conjectured that1 a r X i v : . [ m a t h . C O ] O c t ( C n , K m ) = ( n − m −
1) + 1 for all n ≥ m ≥
3, except n = m = 3 [10, 8].Note that the lower bound is easy: ( m −
1) vertex-disjoint copies of K n − providesa witness for R ( C n , K m ) > ( n − m − m = 8 being the current smallest open case.This work involves fixed n = 4, that is, the case of avoiding the quadrilateral C in the first color. The currently best known asymptotic bounds for R ( C , K m ) are asfollows. Theorem 1 ([23, 2]) . There exist positive constants c and c such that c (cid:18) m log m (cid:19) ≤ R ( C , K m ) ≤ c (cid:18) m log m (cid:19) . The lower bound was obtained by Spencer in 1977 [23] using the probabilistic method.The upper bound was published by Caro, Li, Rousseau, and Zhang in 2000 [2], whoin turn gave credit to an unpublished work by Szemer´edi. The main challenge isdetermining whether R ( C , K n ) < n − (cid:15) for some (cid:15) >
0, a question posed by Erd˝os in1981 [7].Prior to this work, the exact values for R ( C , K m ) were known for 3 ≤ m ≤ R ( C , K ) = 30 and R ( C , K ) = 36.The known values and bounds, including our new results, are gathered in Table 1. m R ( C , K m ) Year References3 7 1971 [3]4 10 1972 [4]5 14 1977 [5]6 18 1987/1977 [9]/[21]7 22 2002/1997 [19]/[12]8 26 2002 [19]9 30 this work10 3611 39–4412 42–53 Table 1: Known values and bounds for R ( C , K m ).Double references correspond to lower and upper bounds. The value of R ( C , K ) and bounds 21 ≤ R ( C , K ) ≤
22 were presented byJayawardene and Rousseau in [12, 13]. The numbers R ( C , K ), R ( C , K ) and thebounds 30 ≤ R ( C , K ) ≤
33, 34 ≤ R ( C , K ) ≤
40 were given by Radziszowskiand Tse in [19]. Further upper bound improvements to 32 and 39 for R ( C , K ) and2 n, C ) 3 4 6 7 9 11 13 16 18 21 n
13 14 15 16 17 18 19 20 21 22ex( n, C ) 24 27 30 33 36 39 42 46 50 52 n
23 24 25 26 27 28 29 30 31 32ex( n, C ) 56 59 63 67 71 76 80 85 90 92 Table 2: Known values for ex( n, C ) [6, 27, 22]. R ( C , K ), respectively, were presented in [26].For graph G , V ( G ) is the vertex set; E ( G ) is the edge set; N G ( v ) is the neighbor-hood of v ∈ V ( G ); deg G ( v ) is | N G ( v ) | ; δ ( G ) is the minimum degree; and α ( G ) is theindependence number. The computations and algorithms used in this work are similar to those described in[19]. Comparable methods have been used to find other Ramsey numbers, such as in[17, 11].The main idea behind the computations is to enumerate the sets R ( C , K m ).If R ( C , K m ; n ) (cid:54) = ∅ , then R ( C , K m ) > n , and if R ( C , K m ; n + 1) = ∅ , then R ( C , K m ) ≤ n + 1. The latter is usually accomplished by extending R ( C , K m ; t )to graphs in sets with higher m and/or t . Two methods used to achieve this aredescribed in the next section.Some special properties of C -free graphs proved useful during our computations.One such property involves an extremal Tur´an-type problem involving C -free graphs.Let ex( n, C ) be the maximum number of edges of an n -vertex C -free graph. Thesenumbers have been studied extensively both theoretically and computationally (cf.[1]). The values of ex( n, C ) for 1 ≤ n ≤
32 are known [6, 27, 22] and they aredisplayed in Table 2.
Lemma 1 ([4, 1]) . If a C -free graph has n vertices, e edges, and minimum degree δ ,then δ − δ + 1 ≤ n and e < n (1 + √ n − . .2 Methods Our enumeration of various classes of ( C , K m )-graphs uses two computational meth-ods, VertexExtend and
Glue , described below.
VertexExtend
This algorithm extends a ( C , K m ; n )-graph G to all possible ( C , K m ; n + 1)-graphs G (cid:48) containing G by attaching a new vertex v to all feasible neighborhoods in G . By feasible, we mean that the additional edges do not create a C while also pre-serving α ( G (cid:48) ) < m . If complexity of computations is ignored, then full enumerationof R ( C , K m ; n + 1) can clearly be obtained from R ( C , K m ; n ) with this method. Glue
The second method, called the
Glue algorithm, constructs R ( C , K m ; n + δ + 1)from R ( C , K m − ; n ), where δ is the minimum degree of the new graphs. For a( C , K m ; n + δ + 1)-graph G , let v ∈ V ( G ) be such that deg G ( v ) = δ ( G ), and let X be the subgraph induced by N G ( v ); X must be a ( P , K m ; δ )-graph. Note that such agraph must be of the form sK ∪ tK , where 2 s + t = δ and s + t < m . Let Y be theinduced subgraph of V ( G ) \ ( X ∪ { v } ); Y must be a ( C , K m − ; n )-graph. If we know R ( C , K m − ; n ), we can find all graphs in R ( C , K m ; n + δ + 1) by considering howeach vertex x ∈ X can be connected to the vertices of Y . We call each neighborhood N ( x ) ∩ V ( Y ) the cone of x , denoted c ( x ). We say that the cone c ( x ) is feasible if:1. c ( x ) does not contain two endpoints of any P in Y .2. For distinct x , x ∈ V ( X ), c ( x ) ∩ c ( x ) = ∅ .3. For each edge { x , x } ∈ E ( X ), there is no y ∈ c ( x ) and y ∈ c ( x ) such that { y , y } ∈ E ( Y ).4. For each subgraph induced by X (cid:48) ⊆ X and Y (cid:48) induced by V ( Y ) \ (cid:83) x ∈ X (cid:48) c ( x ), α ( X (cid:48) ) + α ( Y (cid:48) ) < m .Conditions 1, 2, and 3 prevent C ’s, while condition 4 prevents independent sets oforder m . Figure 1 presents the main idea of Glue . Two separate implementations of
VertexExtend and
Glue were used in order tocorroborate the correctness of the results. In all cases where both implementationswere used, the results agreed. We list the details of this agreement in the Appendix.The rules for gluing ( C , K m )-graphs described in Section 2.2 allowed for a muchneeded speedup in computations. In most cases, it was beneficial to preprocess the Y graphs before gluing, storing information about the feasibility of the cones. For4 δ ( P , K m ; δ )-graph ( C , K m − ; t )-graph Figure 1: Gluing to a ( C , K m ; δ + t + 1)-graph. example, all subsets of vertices containing endpoints of a P were removed from thelist of feasible cones. Speed was greatly increased by precomputing the independencenumber α ( Y (cid:48) ) of each subgraph, which was critical for efficient testing of condition4. This proved to be a bottleneck of the computations, and multiple strategies andimplementations were tested. The most efficient algorithm implemented was basedon Algorithm 1: Precomputing independence number , described in [11]. All data wasstored in arrays of size 2 n , where the integer index of the array represented the bit-setof the vertices of the subgraph.Two isomorphism testing tools were used in our implementations. The first im-plemented an algorithm described by William Kocay [15]. The other made use of thewell-known software nauty by Brendan McKay [16]. First, we obtained a full enumeration of R ( C , K ). This was significant, as the sameenumeration was computationally infeasible when these methods were attempted in2002 [19]. R ( C , K ) was first obtained using VertexExtend . The same resultswere obtained when gluing from R ( C , K ). For more information on these andsimilar consistency checks, see the Appendix. The statistics of R ( C , K ) by vertexand edge counts are displayed in Tables 3 and 4. The cases of counts found in [19]agree with ours.Once R ( C , K ) was obtained, we were able to construct R ( C , K ; n ) for n equalto 23, 24, and 25. The gluing of R ( C , K ; 23) turned out to be the most computation-ally expensive, as there are 353015495 such graphs, but this was needed in order toextend them further to R ( C , K ; 29). The counts for R ( C , K ; 23) are displayed bysize and minimum degree in Table 5. Statistics for R ( C , K ; 24) and R ( C , K ; 25)are gathered in Table 6. Our computations found that no ( C , K )-graph exists with5 e R ( C , K ; n ), 7 ≤ n ≤ n < C -free graphs.
16 17 18 19 20 21 e
14 115 516 23 117 116 318 644 11 119 3602 51 120 19588 251 321 97521 1311 1222 423964 6805 4523 1543985 33476 19824 4434855 149441 90825 9068568 585687 404526 11612126 1964782 1697127 8299450 5448131 6446228 3016205 11583843 21983129 511367 16465694 672324 130 37318 13277929 1813931 1831 1167 5287770 4096321 23332 26 938464 6953952 239933 2 68369 7533349 1747434 2018 4275886 8378635 35 1064229 26109336 1 102512 52055137 3512 605219 138 53 328849 1239 1 64919 12640 4132 99941 107 361142 4 376243 89744 5345 2 146 2Total 39070533 55814073 26822547 1888785 9463 3Table 4: Statistics for R ( C , K ; n ), 16 ≤ n ≤ e
40 1 141 13 1342 201 20143 3055 108 316344 36884 8517 4540145 302179 260678 56285746 1 1449548 3502385 83 495201747 6 3662039 23059729 35368 2675714248 29 4576213 75076644 1563123 8121600949 53 2716695 110589375 11348103 12465422650 27 744258 66302337 19535975 8658259751 3 95358 15327155 9727032 2514954852 5827 1352590 1588719 294713653 164 47152 94684 14200054 6 732 2404 314255 4 37 4156 1 1Total 119 13592441 295527406 43895529 353015495
Table 5: Size vs minimum degree of graphs in R ( C , K ; 23).All such graphs with δ = 4 were used with Glue to find ( C , K ; 29)-graphs. minimum degree 5. R ( C , K ) We constructed the sets R ( C , K ; 29) and R ( C , K ; 30) with the Glue algorithm.Since R ( C , K ) = 26, any ( C , K ; 29)-graph has minimum degree 3, 4, or 5 and canbe obtained from R ( C , K ; n ) for n = 25 , ,
23 by
Glue . Note that the minimumdegree of a ( C , K ; 23)-graph must be 4 in order to glue to a graph of minimumdegree 5. This restriction improved the speed of computation, as there is a largenumber of ( C , K ; 23)-graphs to consider. Statistics for R ( C , K ; 29) are found inTable 7.Similarly, any ( C , K ; 30)-graph has minimum degree 4 or 5, and can be obtainedfrom R ( C , K ; 25) or R ( C , K ; 24), respectively, via Glue . No ( C , K ; 30)-graphswere found, resulting in the following theorem. Theorem 2. R ( C , K ) = 30 .
24 25 e
48 149 650 4851 39452 313353 2111654 6064655 5794456 1886357 210258 96 259 4 1060 1561 9Total 164353 36
Table 6: Statistics for R ( C , K ; n ), n = 24 , C , K ; m )-graphs for m ≥ δ e
70 1 1 271 8 5 1372 12 11 2373 18 33 1 5274 10 64 7 8175 49 9 5876 19 7 2677 6 4 1078 2 2Total 49 188 30 267
Table 7: Size vs minimum degree of graphs in R ( C , K ; 29).These graphs were used to show that no ( C , K ; 36)-graph exists. .2 R ( C , K ) Theorem 3. R ( C , K ) = 36 .Proof. We have found two 6-regular ( C , K ; 35)-graphs H and H , establishing thelower bound. The orbits of H are depicted in Figure 2 and its adjacency matrix ispresented in Figure 3.In order to prove R ( C , K ) ≤
36, it is necessary to show that no ( C , K ; 36)-graph exists. As R ( C , K ) = 30, from Lemma 1, a ( C , K ; 36)-graph has minimumdegree at most 6 and can be obtained from gluing a ( C , K ; 29)-graph. Gluing all of R ( C , K ; 29) resulted in finding no such graphs.The automorphism group Aut( H ) has order 24 and its action on V ( H ) has fourorbits of 24, 6, 4, and 1 vertices, respectively. The automorphism group Aut( H ) hasorder 40 and its action on V ( H ) has three orbits of 20, 10, and 5 vertices. Bothgraphs H and H have 105 edges and 35 triangles. Each vertex is on three triangles,that is, each neighborhood is the union of three K graphs. Both graphs are alsobicritical: removing any edge produces an independent set of order 10, and addingany edge produces a C .Interestingly, no ( C , K ; n )-graphs for n = 34 ,
35 were obtained by gluing from R ( C , K ; 29). ( a ) ( b ) ( c ) ( d ) Figure 2: The four orbits of Aut( H ). Parts ( b ) and ( c ) are connected by 24edges, as well as ( c ) and ( d ). Theorem 4. ≤ R ( C , K ) ≤ .Proof. The lower bound is obtained by construction. A ( C , K ; 38)-graph can easilybe obtained by adding a triangle to H or H .10 b c da b c d Figure 3: Adjacency matrix of H separated by the orbits of Aut( H ). If a ( C , K ; 44)-graph G exists, then from Lemma 1 it follows that G must haveminimum degree at most 7. Such a graph can be obtained by applying Glue toa ( C , K ; 36)-graph. However, since R ( C , K ) = 36, no such graph exists, andtherefore G does not exist as well. Theorem 5. ≤ R ( C , K ) ≤ .Proof. The lower bound is obtained similarly as before, by adding a triangle to the( C , K ; 38)-graphs of Theorem 4.As R ( C , K ) ≤
44, any ( C , K )-graph can be obtained by applying Glue to a ( C , K )-graph with order at most 43. From Lemma 1, such a graph musthave a minimum degree of at most 7, and therefore an order of at most 51. Thus, R ( C , K ) ≤
52. 11
Acknowledgments
This research was done using resources provided by the Open Science Grid, which issupported by the National Science Foundation and the U.S. Department of Energy’sOffice of Science. We owe many thanks to Mats Rynge for his guidance throughoutour use of the OSG, and to Gurcharan Khanna and Research Computing at RIT fortheir valuable and helpful support.
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Glue and
VertexExtend were devel-oped by the authors. The data was generated independently by both implemen-tations, and all results agreed. The only data that was not produced by bothwas the full enumeration of R ( C , K ; 23) and R ( C , K ; 29), as this required themost computational resources. However, a partial set of R ( C , K ; 23), namelywhen δ = 1 ,
2, was verified.2. Both implementations were used to generate all graphs in R ( C , K t ) for 4 ≤ t ≤
7. The results agreed, and gave the same counts as those found in [19].3. For every ( C , K ; 23)-graph, we removed an edge if it did not increase theindependence number to 8, therefore producing a different ( C , K ; 23)-graph.Every graph found this way was already included in the original set. For exam-ple, when going from size 51 to 50, 65059062 of the 86582597 graphs ( ≈ C , K )-graph with 24 and 25 vertices, every vertex was removed,creating a ( C , K )-graph with 23 and 24 vertices, respectively. Every graphproduced was already included in the set obtained earlier.5. Tests 3 and 4 were performed on other sets of graphs, including R ( C , K ; 29).Like before, all graphs obtained this way had already been found.6. We extended R ( C , K ; 29) to R ( C , K ; 30) via VertexExtend and also ob-tained R ( C , K ; 30) = ∅ . Similarly, we extended the ( C , K ; 35)-graphs H and H from Theorem 3 and no ( C , K ; 36)-graphs were found.7. All ( C , K m )-graphs were independently verified to not contain a C or inde-pendent set of order m using the software sagesage