Defective Ramsey Numbers Tınaz Ekim Boğaziçi University Istanbul, Turkey Joint work with J. Gimbel and A. Akdemir

Overview Defective Ramsey numbers Defective cocolorings Definitions Tableaux with best known LB and UB Defective cocolorings Parameter ck(m) LB and UB on ck(m): generalized Straight’s formula Computation of c0(4) , c1(3), c2(2) by efficient graph generation methods CanaDAM - June 2013

Ramsey Numbers R(a,b) is the minimum integer n such that all graphs with n vertices have either an a-clique or a b-independent set as an induced subgraph a,b 1 2 3 4 5 6 7 8 9 10 14 18 23 28 36 40–43 25 35–41 49–61 56–84 73–115 92–149 43–49 58–87 80–143 101–216 125–316 143–442 102–165 113–298 127–495 169–780 179–1171 205–540 216–1031 233–1713 289–2826 282–1870 317–3583 ≤ 6090 565–6588 580–12677 798–23556 CanaDAM - June 2013

k-dense and k-sparse sets A set S of vertices in G is k-sparse if S induces a graph with maximum degree at most k. k-defective coloring Ex: 2-sparse A set S is k-dense if S induces a k-sparse graph in the complement of G. Ex: 2-dense k-sparse or k-dense k-defective set CanaDAM - June 2013

Defective Ramsey Numbers Rk(a, b) : min integer n such that all graphs of order n contain either a k-dense a-set or a k-sparse b-set. Rk(a, b) R(a, b) Cockayne and Mynhardt , 1999 (1-dependent R. N.): R1(3,j)=j ; R1(4, 4) =6 ; R1(4, 5) =9 ; R1(4, 6) = 11 ; R1(4, 7) = 16 ; R1(4, 8) = 17 ; R1 (5, 5) = 15 E. and Gimbel, 2011  R2(5, 5) =7 Chappell and Gimbel, 2011  R2(5, 6) =8 computer many more values with all extremal graphs CanaDAM - June 2013

LB and UB on Rk(a, b) Chappell and Gimbel, 2011: CanaDAM - June 2013

Best LB and UB - 1 Recursively compute best known LB and UB on Rk(a, b) Improve the LB, if possible, by a random graph generator ( a graph of order LB with no k-dense a-set or k-sparse b-set  LB LB+1) CanaDAM - June 2013

Best LB and UB - 2 CanaDAM - June 2013

Best LB and UB - 3 CanaDAM - June 2013

Best LB and UB - 4 CanaDAM - June 2013

Best LB and UB No time limit restriction to enlarge the tables in terms of a, b and k. But the improvement algo stops after 8 hours. There are in total 55 improvements. Highest improvement is obtained for R2(10,10) by increasing LB by 9. Links of Rk(i, j) with defective cocolorings CanaDAM - June 2013

Defective Cocolorings z(G): min # of independent sets and cliques partitioning V zk(G): min # of k-defective sets partitioning V ck(m): max n s.t. all n-graphs has a k-defective m-cocoloring (zk (G)  m for all n-graphs G) Straight, 1980: c0(2)=4 c0(3)=8 c0(4){11,12} Every 11-graphs G has z0(G)4 since it contains either a 3-clique or a 3-independent set by R(3,3) = 6 and the remaining graph on 8 vertices needs at most 3 colors as c0(3) = 8. Since R(4,4)=18  13-graph G with no 4-clique and 4-independent set z0(G)=5 CanaDAM - June 2013

Generalized Straight’s formula c1(2)=7 (E. and Gimbel, 2011) R1(4; 4) =6  1 color for 4 vertices Remaining 3 vertices are 1-defective.  c1(2)7 There is an 8-graph with z1(G)=3  c1(2)=7 CanaDAM - June 2013

Applications of Generalized Straigth’s formula (E. and Gimbel, 2011 ) c1(3) 11 (and c1(3) 15 ) R1(4; 4) =6  1 color for 4 vertices c1(2)=7  remaining 7 vertices needs 2 colors Is there a 12-graph s.t. z1 (G)=4? c2(2)  9 (and c2(2)  13) R2(5; 5) =7  1 color for 5 vertices Remaining 4 vertices are 2-defective Is there a 10-graph s.t. z2(G)=3? CanaDAM - June 2013

Computer aided search c1(3) 11 Is there a 12-graph s.t. z1 (G)=4? IDEA: Generate all 12-graphs with 1-defective cochromatic number 4, if any. If the algo returns no graph at all, it means that all 12-graphs has z1 (G)3 then c1(3) 12. Use a random graph generator to generate a 13-graph with z1 (G)=4 (verified by the checks embedded in the previous algo), if any CanaDAM - June 2013

Eliminate before generating! 165.091.172.592 graphs of order 12 + check if 1-defectively 3-cocolorable (245.817 configurations to check for each 12-graph) Generation of 12-graphs with 1-def. cochromatic number 4: No 1-def. 5-set since c1(2)=7 R1(4; 4) =6  1-def. 4-set Remaining 8 vertices  1-def. 4-set Remaining 4 vertices not 1-defective Generate all graphs (if any) s.t. 1-def 4-set + 1-def 4-set + not 1-def 4-set Not containing 1-def 5-set Check if they can be partitioned into three 1-def 4-sets CanaDAM - June 2013

c0(4)=12 (Akdemir and E., 2013) 21 days All 12-graphs are 4-cocolorable  13-graph with z(G)=5 CanaDAM - June 2013

c1(3)=12 (Akdemir and E., 2013) 5 days All 12-graphs are 1-defectively 3-cocolorable  13-graph with z1(G)=4 CanaDAM - June 2013

c2(2)=10 (Akdemir and E., 2013) 7 hours All 10-graphs are 2-defectively 2-cocolorable  11-graph with z2(G)=3 CanaDAM - June 2013

k-defective cocritical graphs CanaDAM - June 2013

Open questions Improve LB for Rk(a, b) Theoretical proofs for c0(4) , c1(3), c2(2) Other exact values for ck(m): c0(5){14,15} 16  c2(3) 22 12  c3(2) 17 Defective Ramsey Numbers and ck(m) in restricted graph classes. CanaDAM - June 2013

Thank you for your attention! CanaDAM - June 2013