We extend Goodman’s result on the cardinality of monochromatic triangles in a 2-colored complete graph to the case of bounding the number of triangles in the first color. We apply it to derive the upper bounds on some non-diagonal Ramsey numbers. In particular we show that R(K4 − e,K8) \u3c= 45

Lin, Andrew

Mackey, John

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Counting triangles in some Ramsey graphs

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Rochester Institute of TechnologyRIT Scholar WorksArticles2009Counting triangles in some Ramsey graphsAndrew LinJohn MackeyFollow this and additional works at: http://scholarworks.rit.edu/articleThis Technical Report 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 CitationLin, Andrew and Mackey, John, "Counting triangles in some Ramsey graphs" (2009). Accessed fromhttp://scholarworks.rit.edu/article/983Counting Trianglesin Some Ramsey GraphsAndrew LinDepartment of Computer ScienceRochester Institute of TechnologyRochester, NY 14623, USAapl8378@cs.rit.eduJohn MackeyDepartment of Mathematical ScienceCarnegie Mellon UniversityPittsburgh, PA 15213, USAjmackey@andrew.cmu.eduAbstractWe extend Goodman’s result on the cardinality of monochromatictriangles in a 2-colored complete graph to the case of bounding thenumber of triangles in the first color. We apply it to derive the upperbounds on some non-diagonal Ramsey numbers. In particular we showthat R(K4 − e,K8) ≤ 45.AMS Subject Classification: 05C5511 IntroductionFor any two simple graphs H1 and H2, the Ramsey number R(H1, H2) is theleast positive integer n such that for every n-vertex graph G, either H1 isa subgraph of G or H2 is a subgraph of G. A graph G such that neitherH1 is a subgraph of G nor H2 a subgraph of G is said to be (H1, H2)-good.An (H1, H2)-good graph is (H1, H2)-critical if there exists no (H1, H2)-goodgraph with more vertices. The standard Ramsey case involving two completegraphs is written as R(m,n) = R(Km, Kn). Ramsey numbers can also begeneralized to cases involving more than two colors and hypergraphs [Rad06].Such cases are not covered here.To this day, only nine non-trivial exact values of classical Ramsey num-bers are known. [Rad06]. In practice, often a lower bound for R(m,n) isfound by producing a graph which contains no cliques of order m and noindependent sets of order n, or by a proof demonstrating that the probabil-ity of the existence of such graphs is non-zero. An upper bound for R(m,n)is established by proving that every graph on a given number of verticescontains a clique of order m or an independent set of order n.2 Related WorkGoodman [Goo59] proved that the number of monochromatic triangles in asimple graph is entirely determined by the degrees of each vertex. He showedthat for G = (V,E),∑v∈V(d(v)3)+(v − 1− d(v)3)= (v − 3)T (1)where T is the cardinality of K3 and I3(= K3) subgraphs in G. Several gen-eralizations have been proposed afterwards, including that of Giraud, whogave a lower bound for the cardinality of monochromatic K4 [Gir73]. Mackey[Mac94] later demonstrated how to apply Giraud’s method to some cases ofdiagonal Ramsey numbers. Most notably, Mackey demonstrated the newupper bounds of 165, 540, 1870, 6625, 23854 for the Ramsey numbers R(6, 6)2through R(10, 10), respectively.Our main result is a further generalization of this idea to allow one tobound the number of monochromatic K3 or I3 independently of each other.We demonstrate how to improve the upper bounds of some off-diagonal Ram-sey numbers using this result.3 Identities Involving Vertex DegreesThe edges of G will be denoted by E and the non-edges E. We let gm,ndenote the number of m-vertex subgraphs of G which are isomorphic to thegraph labeled Gm,n in figure 1.r rrG3,0HHHr rrG3,1HHHr rrG3,2HHHr rrG3,3rrrrG4,0rrrrG4,1rrrrG4,2rrrrG4,3@@@rrrrG4,4rrrrG4,5   rrrrG4,6rrrrG4,7@@@rrrrG4,8   rrrrG4,9   @@@rrrrG4,10Figure 1. Labeling of graphs on 3 and 4 vertices.3For each edge or non-edge e = {a, b} and i = 0, 1, or 2, we let yi(e) tobe defined as the number of vertices not contained in e which are connectedto exactly i members of e. We also let ya,1(e) be the number of verticesnot contained in e which are connected to a and not connected to b, soya,1(e) + yb,1(e) = y1(e). Although the latter definition depends on the la-beling of e, there will be no ambiguity in expressions which is symmetric inya,1(e) and yb,1(e).In a simple graph, the sum of the degrees of each vertex is twice the num-ber of edges. By simple subgraph counting, we can generate three identitiesinvolving quadratic sums of vertex degrees as follows.3g3,3 + g3,2 =∑v∈V(d(v)2)=12∑v∈Vd(v)2 − |E|2g3,2 + 2g3,1 =∑v∈Vd(v)[|V | − 1− d(v)] = −∑v∈Vd(v)2 + 2(|V | − 1)|E|g3,1 + 3g3,0 =∑v∈V(|V | − 1− d(v)2)=12∑v∈Vd(v)2 − (2|V | − 3)|E|+ |V |(|V | − 1)(|V | − 2)2(2)The identities in in (2) allow one to write g3,2, g3,1, and g3,0 in terms offour quantities: |V |, |E|, ∑v∈V d(v)2, and g3,3, as follows.g3,2 = −3g3,3 + 12∑v∈Vd(v)2 − |E|g3,1 = 3g3,3 −∑v∈Vd(v)2 + |V ||E|g3,0 = −g3,3 + 12∑v∈Vd(v)2 − (|V | − 1)|E|+ |V |(|V | − 1)(|V | − 2)6(3)Identity (4) can be verified by looking at subgraphs of four vertices.4∑e∈E[(ya,1(e)2)+(yb,1(e)2)]+∑e∈E[(ya,1(e)2)+(yb,1(e)2)]= g4,2 + 3g4,4 + 3g4,6 + g4,8=∑e∈Ey2(e)y0(e) +∑e∈Ey2(e)y0(e) (4)By applying the Cauchy-Schwarz inequality to the LHS of (4), we have∑e∈E[(ya,1(e)2)+(yb,1(e)2)]=12∑e∈E[(ya,1(e))2 + (yb,1(e))2]− 12∑e∈Ey1(e)≥ 14∑e∈E[y1(e)]2 − 12∑e∈Ey1(e)≥ 14|E|(∑e∈Ey1(e))2 − 12∑e∈Ey1(e)=g23,2|E| − g3,2 (5)Similarly,∑e∈E[(ya,1(e)2)+(yb,1(e)2)]≥ g23,1|E| − g3,1 (6)An upper bound for the RHS of (4) can be found by rearranging andapplying Cauchy-Schwarz as follows.Theorem 1∑e∈Ey2(e)y0(e) ≤ 3(v − 2)g3,3 −9g23,3|E| − 3g3,3LEand∑e∈Ey2(e)y0(e) ≤ 3(v − 2)g3,0 −9g23,0|E| − 3g3,0LE5for any LE ≤ min{y1(e)|e ∈ E} and LE ≤ min{y1(e)|e ∈ E}.Proof:We arrive at these bounds by making a simple substitution and applyingthe Cauchy-Schwarz inequality as follows.∑e∈Ey2(e)y0(e) =∑e∈Ey2(e)[v − 2− y2(e)− y1(e)]= (v − 2)∑e∈Ey2(e)−∑e∈E[y2(e)]2 −∑e∈Ey2(e)y1(e)≤ (v − 2)∑e∈Ey2(e)− [∑e∈E y2(e)]2|E| − LE∑e∈Ey2(e)= 3(v − 2)g3,3 − 9(g3,3)2|E| − 3LEg3,3 (7)for any LE ≤ min{y1(e)|e ∈ E}, and∑e∈Ey2(e)y0(e) =∑e∈Ey0(e)[v − 2− y0(e)− y1(e)]= (v − 2)∑e∈Ey0(e)−∑e∈E[y0(e)]2 −∑e∈Ey0(e)y1(e)≤ (v − 2)∑e∈Ey0(e)− [∑e∈E y0(e)]2|E| − LE∑e∈Ey0(e)= 3(v − 2)g3,0 − 9(g3,0)2|E| − 3LEg3,0 (8)for any LE ≤ min{y1(e)|e ∈ E}. We thus obtain the following inequality among the two and three vertexsubgraph cardinalities and LE and LE.69(g3,3)2 + (g3,2)2|E| +(g3,1)2 + 9(g3,0)2|E| −(v3)− (3v − 7)(g3,3 + g3,0)+ 3LEg3,3 + 3LEg3,0 ≤ 0 (9)Noting the relationships in (2), and then solving for g3,3, we obtain thefollowing bound:g3,3 ≥ 112v(v − 1){ (LE − LE)|E|(2|E| − v2 + v) + (8|E|+ v2 − v)s−16(v − 1)|E|2 + 2v(v − 1)(v − 3)|E| − Φ 12} (10)whereΦ = (LE − LE)2|E|2(2|E| − v2 + v)2+2|E|LE(2|E| − v2 + v)[8s|E|+ v(v − 1)s−16(v − 1)|E|2 + 2v(v − 1)(v − 3)|E|]−2|E|LE(2|E| − v2 + v)[8s|E| − 5v(v − 1)s−16(v − 1)|E|2 + 2v(v − 1)(7v − 9)|E| − 2v2(v − 1)2(v − 2)]+64s2|E|2 − 32v(v − 1)s2|E| − v2(v − 1)2s2−256(v − 1)s|E|3 + 8v(v − 1)(19v − 21)s|E|2−8v2(v − 1)2(v − 2)s|E|+ 256(v − 1)2|E|4−16v(v − 1)2(11v − 13)|E|3 + 4v2(v − 1)3(5v − 9)|E|2,s =∑v∈Vd(v)2.It is important to note that this lower bound depends on only five graphparameters: v, E,∑v∈V d(v)2, LE, and LE.4 Proof that R(K4 − e,K8) ≤ 45It is currently known that R(K4 − e,Kn) = 7, 11, 15, 21 for n = 3, 4, 5, 6,respectively [CH72] [BH81] while the best known bounds for R(K4 − e,K7)7and R(K4 − e,K8) are only ones obtained by general inequalities, such as2.3(a) of [Rad06], and the fact that Ramsey numbers are monotonic increas-ing. We see that R(K4 − e,K7) ≥ R(K4 − e,K7 − e) = 28 [MR91] andR(K4−e,K7) ≤ R(P3, K7)+R(K4−e,K6) = 34. Similarly, R(K4−e,K8) ≤R(K4−e,K7)+R(P3, K8) ≤ 21+13 = 49 and R(K4−e,K8) ≥ R(K4−e,K8−e) ≥ R(K4−e,K7−e)+3 = 31. The latter bound is given by observing thatany (K4 − e,K7 − e)-good graph can be extended to (K4 − e,K8 − e)-goodby adding an isolated K3.Next, we demonstrate how to apply inequality (10) to improve this bound,and show that R(K4 − e,K8) ≤ 45. Our proof is by contradiction.Theorem 2 R(K4 − e,K8) ≤ 45.Proof: Suppose that G = (V,E) is a 45-vertex (K4 − e,K8)-good graph.For each vertex v ∈ V , by the obvious Ramsey constraints of the sub-graphs induced by the vertices either connected to or not connected to v, wesee that 11 ≤ d(v) ≤ 14, since R(K4 − e,K7) ≤ 34 and R(K3 − e,K8) = 15.For each e ∈ E, y2(e) ≤ 1 and ya,1(e) + y0(e) ≤ R(K4 − e,K7)− 1 ≤ 33.Thus, yb,1(e) ≥ 9. In a similar way, ya,1(e) ≥ 9, and thus y1(e) ≥ 18.Similarly, for e ∈ E, y0(e) ≤ R(K4 − e,K6) − 1 ≤ 20, y2(e) + ya,1(e) ≤R(K3 − e,K8) − 1 = 14, and thus y1(e) ≥ 18. Thus, LE = LE = 18 aresuitable values for the above bound, which then becomes:g3,3 ≥ 12970(∑v∈Vd(v)2)|E|+ 112(∑v∈Vd(v)2)− 4135|E|2 + 7|E|− 123760{(64|E|2 − 63360|E| − 3920400)(∑v∈Vd(v)2)2−(11264|E|2 − 14065920|E|+ 2195424000)(∑v∈Vd(v)2)|E|+495616|E|4 − 747141120|E|3 + 247832006400|E|2−24032365632000|E|} 12 (11)8Since 11 ≤ d(v) ≤ 14 for each vertex v ∈ V , we can obtain the bounds248 ≤ |E| ≤ 315. Using the upper bound of d(v) as well as the Cauchy-Schwarz inequality, we obtain 445|E|2 = 4|E|2v≤ ∑v∈V d(v)2 ≤ 28|E|.In a (K4 − e)-free graph, it is clear that for all e ∈ E, y2(e) ≤ 1 andthus 3g3,3 ≤ |E|. However, looping through the above constraints of |E| and∑v∈V d(v)2 and looking at the lower bound of g3,3, as given in (11), we seethat this is not satisfiable. Thus G does not exist and R(K4− e,K8) ≤ 45. 5 ConclusionsThe best known bound for R(K4 − e,K7) is the bound of R(K4 − e,K7) ≤R(K3 − e,K7) + R(K4 − e,K6) = 13 + 21 = 34, using identity 2.3(a) of[Rad06]. Our approach of theorem 2 does not appear to improve it. It isalso not likely to improve any other bounds of the form R(K4 − e,Kn) forn ≥ 9. In a similar fashion, this approach also confirms that R(3, 10) ≤ 43,but does not seem to improve the bound for any cases of R(3, n) for n ≥ 11.This method demonstrates a generalization of Goodman’s bound on thecardinality of 3-vertex cliques to that of off-diagonal cases, and provides ameans to bound some cases of off-diagonal Ramsey numbers. In particular,cases in which the cardinality of 3-vertex cliques is bounded relative to thecardinality of edges, such as those involving the book graphs Bn, are likelyto give way to such approach.References[BH81] R. Bolze and H. Harborth. The Ramsey number R(K4 − x,K5).The Theory and Applications of Graphs, John Wiley & Sons, NewYork, pages 109–116, 1981.[CH72] V. Chva´tal and F. Harary. Generalized Ramsey theory for graphs,iii. Small off-diagonal numbers. Pacific Journal of Mathematics, 41,pages 335–345, 1972.[Gir73] G. Giraud. Une minoration du nombre de quadrangles unicoloreset son application a´ la majoration des nombres de Ramsey binaires-9bicolores. C.R. Acad. Sc. Paris, Se´ries A, 276, pages 1173–1175,1973.[Goo59] A. Goodman. On sets of acquaintances and strangers at any party.Amer. Math. Monthly, Vol. 66, pages 778–783, 1959.[Mac94] J. Mackey. Combinatorial remedies. Ph.D. Thesis, University ofHawaii, 1994.[MR91] J. McNamara and S. Radziszowski. The Ramsey number R(K4 −e,K6 − e) and R(K4 − e,K7 − e). Congressus Numerantium, 81,pages 89–96, 1991.[Rad06] S. Radziszowski. Small Ramsey numbers. The ElectronicJournal of Combinatorics, http://www.combinatorics.org/Surveys/ds1/toc.pdf, DS1, 2006.10

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Counting triangles in some Ramsey graphs

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