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

2. 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

3. 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
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4. 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

5. 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
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6. LB and UB on Rk(a, b)
Chappell and Gimbel, 2011:
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7. 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)
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8. Best LB and UB - 2
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9. Best LB and UB - 3
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10. Best LB and UB - 4
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11. 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

12. 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

13. 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

14. 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

15. 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

16. Eliminate before generating!
graphs of order 12 + check if 1-defectively 3-cocolorable ( 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
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17. c0(4)=12 (Akdemir and E., 2013)
21 days All 12-graphs are 4-cocolorable
 13-graph with z(G)=5
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18. 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

19. c2(2)=10 (Akdemir and E., 2013)
7 hours All 10-graphs are 2-defectively 2-cocolorable
 11-graph with z2(G)=3
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20. k-defective cocritical graphs
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21. 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.
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22. Thank you for your attention!
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