Defective Ramsey Numbers Tnaz Ekim Boazii University Istanbul

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

Overview �Defective Ramsey numbers �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 c 0(4) , c 1(3), c 2(2) by efficient graph generation methods Cana. DAM - June 2013 2

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

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 Cana. DAM - June 2013 4

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. ): R 1(3, j)=j ; R 1(4, 4) =6 ; R 1(4, 5) =9 ; R 1(4, 6) = 11 ; R 1(4, 7) = 16 ; R 1(4, 8) = 17 ; R 1 (5, 5) = 15 �E. and Gimbel, 2011 R 2(5, 5) =7 �Chappell and Gimbel, 2011 R 2(5, 6) =8 computer many more values with all extremal graphs Cana. DAM - June 2013 5

LB and UB on Rk(a, b) Chappell and Gimbel, 2011: Cana. DAM - June 2013 6

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) Cana. DAM - June 2013 7

Best LB and UB - 2 Cana. DAM - June 2013 8

Best LB and UB - 3 Cana. DAM - June 2013 9

Best LB and UB - 4 Cana. DAM - June 2013 10

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 R 2(10, 10) by increasing LB by 9. Ø Links of Rk(i, j) with defective cocolorings Cana. DAM - June 2013 11

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 mcocoloring (zk (G) m for all n-graphs G) �Straight, 1980: c 0(2)=4 c 0(3)=8 c 0(4) {11, 12} Every 11 -graphs G has z 0(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 c 0(3) = 8. Since R(4, 4)=18 13 -graph G with no 4 -clique and 4 independent set z 0(G)=5 Cana. DAM - June 2013 12

Generalized Straight’s formula �c 1(2)=7 (E. and Gimbel, 2011) R 1(4; 4) =6 1 color for 4 vertices Remaining 3 vertices are 1 -defective. c 1(2) 7 There is an 8 -graph with z 1(G)=3 c 1(2)=7 Cana. DAM - June 2013 13

Applications of Generalized Straigth’s formula (E. and Gimbel, 2011 ) �c 1(3) 11 (and c 1(3) 15 ) R 1(4; 4) =6 1 color for 4 vertices c 1(2)=7 remaining 7 vertices needs 2 colors Is there a 12 -graph s. t. z 1 (G)=4? �c 2(2) 9 (and c 2(2) 13) R 2(5; 5) =7 1 color for 5 vertices Remaining 4 vertices are 2 -defective Is there a 10 -graph s. t. z 2(G)=3? Cana. DAM - June 2013 14

Computer aided search �c 1(3) 11 Is there a 12 -graph s. t. z 1 (G)=4? IDEA: 1. Generate all 12 -graphs with 1 -defective cochromatic number 4, if any. 2. If the algo returns no graph at all, it means that all 12 graphs has z 1 (G) 3 then c 1(3) 12. 3. Use a random graph generator to generate a 13 -graph with z 1 (G)=4 (verified by the checks embedded in the previous algo), if any Cana. DAM - June 2013 15

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 c 1(2)=7 R 1(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. Check if they can be � 1 -def 4 -set + not 1 -def 4 -set partitioned into three �Not containing 1 -def 5 -set 1 -def 4 -sets Cana. DAM - June 2013 16

c 0(4)=12 (Akdemir and E. , 2013) � 21 days All 12 -graphs are 4 -cocolorable � 13 -graph with z(G)=5 Cana. DAM - June 2013 17

c 1(3)=12 (Akdemir and E. , 2013) � 5 days All 12 -graphs are 1 -defectively 3 -cocolorable � 13 -graph with z 1(G)=4 Cana. DAM - June 2013 18

c 2(2)=10 (Akdemir and E. , 2013) � 7 hours All 10 -graphs are 2 -defectively 2 cocolorable � 11 -graph with z 2(G)=3 Cana. DAM - June 2013 19

k-defective cocritical graphs Cana. DAM - June 2013 20

Open questions �Improve LB for Rk(a, b) �Theoretical proofs for c 0(4) , c 1(3), c 2(2) �Other exact values for ck(m): � c 0(5) {14, 15} � 16 c 2(3) 22 � 12 c 3(2) 17 �Defective Ramsey Numbers and ck(m) in restricted graph classes. Cana. DAM - June 2013 21

Thank you for your attention! Cana. DAM - June 2013 22