With the help of computer algorithms, we improve the lower bound on the Ramsey multiplicity of K4, and thus show that the exact value of it is equal to 9

Piwakowski, Konrad

Radziszowski, Stanislaw

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The Ramsey multiplicity of K_4

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Rochester Institute of TechnologyRIT Scholar WorksArticles2001The Ramsey multiplicity of K_4Konrad PiwakowskiStanislaw RadziszowskiFollow this and additional works at: http://scholarworks.rit.edu/articleThis Article is brought to you for free and open access by RIT Scholar Works. It has been accepted for inclusion in Articles by an authorizedadministrator of RIT Scholar Works. For more information, please contact ritscholarworks@rit.edu.Recommended CitationArs Combinatoria LX (2001) 131-136The Ramsey Multiplicity of K4Konrad Piwakowski∗ Stanisław P. RadziszowskiDepartment of Foundations of Informatics Department of Computer ScienceTechnical University of Gdan´sk Rochester Institute of Technology80-952 Gdan´sk, Poland Rochester, NY 14623, USAconi@pg.gda.pl spr@cs.rit.eduAbstractWith the help of computer algorithms, we improve the lower boundon the Ramsey multiplicity of K4, and thus show that the exact value ofit is equal to 9.The Ramsey multiplicity M(G) of a graph G is defined as the smallest number ofmonochromatic copies ofG in any two-coloring of edges of KR(G), whereR(G) is theRamsey number of G, i.e. the smallest integer n such that any two-coloring of edges ofKn contains monochromatic copy of G.The study of Ramsey multiplicity was initiated in 1974 by Harary and Prins [3] whodeterminedM(G) for all graphsG of order four or less, except forK4 andK4−e. Thevalue of M(K4 − e) was later determined by Schwenk (cited in [2]). The upper boundM(K4) ≤ 12 was given in 1980 by Jacobson [4], and in 1988 Exoo [1] improved it by3. The only nontrivial lower boundM(K4) ≥ 4 was recently presented by Olpp [7]. Inthis paper we improve this lower bound and thus show that M(K4) = 9.In the sequel, any two-coloring of the edges of Kn containing k monochromaticcopies of K4 is called an (n, k)-coloring. We say that two colorings are isomorphicif the graphs induced by the edges in the first color are isomorphic. Define M(n, k)to be set of all (n, k)-colorings. For a given (n, k)-coloring C let H(C) denote thehypergraph formed by monochromatic copies of K4 in C. Let us defineMd(n, k) tobe the subset of all colorings C ∈ M(n, k) such that the maximal vertex degree inH(C) is equal to d.Our computational approach was to generate all nonisomorphic (18, k)-coloringsfor 4 ≤ k ≤ 8, by iterating an exhaustive enumeration of all possible one vertexextensions of (n − 1, k −m)-colorings to (n, k)-colorings, for m ≥ 0. Let us defineE(n−1, k−m,m) to be the subset of all colorings fromM(n, k) which are one vertexextensions of some coloring fromM(n− 1, k −m).∗ Supported by the State Committee for Scientific Research KBN and academic computer center TASK.1Let V (C) denote the set of vertices of coloring C. For each subset W ⊆ V (C)let N3(W ) denote the sum of the number of triangles in the first color induced byW in C and the number of triangles in the second color induced by V (C)\ W in C.The following algorithm was used to perform the exhaustive search for all one vertexextensions E(n− 1, k −m,m):Algorithm 1Step 1: Initialize output set Out = ∅.Step 2: For each coloring C fromM(n− 1, k −m) execute steps 3, 4, 5.Step 3: For each subset W ⊆ V (C) such that N3(W ) = m execute steps 4, 5.Step 4: Create copy D of coloring C.Step 5: Add a new vertex v to coloring D. For each w in V (C), assign color 1to edge {v, w}, if w ∈ W , and assign color 2 to edge {v, w}, if w ∈ V (C)\W .Add this coloring to Out.Step 6: Remove isomorphic copies from Out.The following lemmas describe computational steps we followed in order to gen-erate colorings of higher orders. As the initial step, we generated the setM(11, 0) byfiltering out (11, 0) colorings from all nonisomorphic graphs of order 11 (which weretreated as two-colorings of K11). The proofs of the lemmas are straightforward byconsidering degree sequences of all possible hypergraphsH(C) in each case.Lemma 1M(n, 0) = E(n− 1, 0, 0), for n ≥ 2,M(n, k) =k−1⋃j=0E(n− 1, j, k − j), for k ≥ 1, and n ≥ 2,M(16, 4) \M1(16, 4) =2⋃j=0E(15, j, 4− j).All the setsM(n, k), for 12 ≤ n ≤ 16, and 0 ≤ k ≤ 3 such that there is a nonemptyentry for n, k in Table 1, were obtained by running Algorithm 1 for the terms on theright hand side of the first two rules in Lemma 1. For example,M(16, 3) was obtainedby extending colorings fromM(15, 0),M(15, 1) andM(15, 2).The last identity in Lemma 1 describes the way of enumerating all (16, 4)-coloringsexcept those whose monochromatic copies of K4 are vertex disjoint (denoted byM1(16, 4)). Unfortunately, there is a frightfully large number of (13, 1) and (14, 2)colorings, and we were not able to complete the sequence of extensionsM(12, 0) →M(13, 1) → M(14, 2) → M(15, 3) → M1(16, 4). Instead, in order to generateM1(16, 4), we used the following approach:2Algorithm 2Step 1: Generate the set of all 2-colorings of order 8 and extract from itM1(8, 2).Step 2: GenerateM1(12, 3) by exhaustively extending by 4 vertices all color-ings inM1(8, 2).Step 3: GenerateM1(16, 4) by exhaustively extending by 4 vertices all color-ings inM1(12, 3).In steps 2 and 3 exactly one new monochromatic K4 is induced by 4 new vertices.As a result of the above algorithm we obtained 468 nonisomorphic (16, 4) colorings.The following lemma, together with Lemma 1, describes the remaining computa-tional steps.Lemma 2M(n, k) =k−2⋃j=0E(n− 1, j, k − j), for k ≥ 5, and n ≤ 19.Using Algorithm 1 and Lemma 1 for k ≤ 4, and Lemma 2 for k ≥ 5, we were ableto generateM(17, 0), ...,M(17, 6) andM(18, 0), ...,M(18, 8).Table 1. The number of nonisomorphic (n, k)-colorings.n \ k 0 1 2 3 4 5 6 7 811 54635612 144916613 118423114 130816 614482015 640 50726 249113616 2 28 382 19806 88844017 1 0 0 2 18 202 575718 0 0 0 0 0 0 0 0 0Table 1 presents the number of nonisomorphic (n, k)-colorings for all n and k,which were enumerated during our computations. The emptiness of the setsM(18, 0), ...,M(18, 8) implies the main theorem:Theorem 1 M(K4) = 9.It is a natural goal to enumerate the setM(18, 9). Continuing our approach wouldrequire obtaining the whole set of coloringsM(17, 7). The latter was unfeasible, andwe were able to enumerate only the setM(17, 6).3Figure 1. The new (18,9)-coloringSince, similar to the previous lemmas, we easily haveM(18, 9) \M2(18, 9) =6⋃j=0E(17, j, 9− j),we enumerated all (18, 9)-colorings such that not every vertex belongs to exactly twomonochromatic copies of K4. There are 4 such colorings, where two of them comefrom the other two by exchanging the colors. Of the two essentially different col-orings, one was presented in [1] and the other is presented in Figure 1, where onlythe edges in one color are shown. There are seven K4 in the first color induced byvertex sets: {A,B,D,E}, {B,C,E, F}, {D,E,G,H}, {E,F,H, I}, {J,C,G,M},{K,A, I, L}, {J,K,L,M} and two K4 in the second color induced by {B,O, P,H}4and {N,D, F,Q}. Notice that the labels Q and R in the Figure 2 in [1] are mistakenlyswitched. It results in serious complications with decoding the (18, 9)-coloring by thereader.The question about contents of the setM2(18, 9) remains open; however we con-jecture that it is empty.Three powerful programs, nauty, makeg, and autoson, implemented by BrendanMcKay [5] were used in our work. All the algorithms specific for this project werewritten independently by both authors, and then a very large number of intermediateand final graphs were tested for isomorphism between the two implementations. More-over, the cardinalities of all setsM(n, 0), for n = 11, ..., 18 agreed with the previousenumeration described in [6].References[1] G. Exoo, Some Constructions Related to Ramsey Multiplicity, Ars. Comb. 26(1988) 233-242.[2] F. Harary, A survey of Generalized Ramsey Theory, in: Graphs and Combinatorics(R. Bari and F. Harary, eds.), Lecture Notes in Mathematics 406, Springer (1974)10-17.[3] F. Harary, G. Prins, Generalized Ramsey Theory for Graphs IV, the Ramsey Mul-tiplicity of a Graph, Networks 4 (1974) 163-173.[4] M. S. Jacobson, A note on Ramsey multiplicity, Discrete Math. 29 (1980) 201-203.[5] B. D. McKay, Nauty User’s Guide (version 1.5), Technical report TR-CS-90-02,Computer Science Department, Australian National University, 1990. WWW ref-erence: http://cs.anu.edu.au/people/bdm/.[6] B. D. McKay, S. P. Radziszowski, A New Upper Bound for the Ramsey NumberR(5, 5), Australasian Journal of Combinatorics, 5 (1992), 13-20.[7] D. Olpp, A Lower Bound for the Ramsey Multiplicity ofK4, Ars. Comb. 49 (1998)289-295.5

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The Ramsey multiplicity of K_4

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