1. The chromatic gap and its extremes
The chromatic gap and its extremes
András Sebő, CNRS, G-SCOP,Grenoble
with
András Gyárfás MTA SZTAKI, Budapest,
Nicolas Trotignon, CNRS, LIAFA, Paris

2. THE PROBLEM
What is the maximum of the difference chromatic number – max clique that can be reached by graphs on at most n vertices ?

3. (G) := min cover by cliques = chromatic number of complement
(G) := max stable (independent) set of vertices (no induced edge)
gap (G) := (G) - (G)  0
This is the (integer linear programming) duality gap of a related linear program

4. Perfect, critical, extremal
Perfect : gap = 0  induced subgraph
Gap is not necessarily monotonous:
gap-critical: G, gap(H)<gap(G)  induced H
Minimal imperfect : gap-critical with gap = 1.
t-extremal graph : of min order with gap t
t-extremal  gap-critical  factorcritical?
= 3
 = 3
gap=0
 = 2
gap=1

5. Critique, pas critique ?
G facteur-critique ?
C9 ?
C93 ? G

6. s(t) = smallest order of a graph of gap t
gap(n)=max {gap(G) : G has n vertices}
gap-extremal ?
gap(i)=0 i 4,
gap-critical !
...
gap(5)=1, gap(10)=2, … ???
s(t) = smallest order of a graph of gap t
s(1)=5, s(2)=10, … , s(t)= 5t ?

7. … s(3)=15 ? Counterexample: n=13, =2, =4, =7 gap = 3 s(3)= 13
R13= (3,5)-Ramsey
n=13, =2, =4,
=7 gap = 3
s(3)= 13

8. Ramsey: number, graph
Ramsey number : R(,):=the smallest integer s.t.:
every graph on at least R(,) vertices
has either an -clique or an -stable set.
(Ramsey theorem: it is finite.)
Ramsey graph : Graph on R(,) – 1 vertices having neither an -clique nor an -stable set.
Example : R(3,3) =6; (3, 3) Ramsey graph
R(3,2), R(3,3), … : 2, 6, 9, 14, 18, 23, 28, 36, ?
R(4,4), R(4,5) : 18, 25, ?

9. gap(G) + 1  gap(G-Q)  gap(G) - 1
EASY FACTS
Fact 1 : If C1 , … , Ck the components of G,
gap(G) = gap(C1) + … + gap (Ck)
G is gap-critical  all of its components are.
Fact 2 : If G is a graph and Q a clique :
(G)  (G-Q)  (G)-1, (G)  (G-Q)  (G)-1
gap(G) + 1  gap(G-Q)  gap(G) - 1
Fact 3 : If a t-extremal graph has a k-clique,
s(t)  s(t-1) + k

10. How big should  be for a big gap ?
gap=  -    n / 
 large ~  small ~  large (Ramsey)
How does gap =  -  change if  varies ?
Conjecture: t-extremal graphs are -free (=2).
gap2 (n) : max gap of a triangle-free on n vertices
s2 (t) : min order of a triangle-free of gap t.
Theorem : s(t) = 2t +  (sqrt (t log t ) ) = s2 (t)
gap(n)=gap2 (n), s(t)=s2 (t) for all n, t

11. Inverse Ramsey (n) := min {(G): G triangle-free of order n}
If R(3,  )  n < R(3,  + 1 ) ,then (n) = 
For instance : (5) = 2 , (6) = 3
Fact: If n = n1 + n2 , then (n)  (n1) + (n2)
If ‘’=‘’ n is Ramsey-perfect; (10)=(5) + (5)
Conjecture: This is the only nontriv example

12. A version of the main result
Theorem: n / 2 - (n)  gap(n)  n / 2 - (n)+3
and with the exception of Ramsey numbers +
1, … , 14, there is equality with the lower bound
The extremal graphs are triangle-free.

13. The proof relies on two lucky facts :
(1) gap2 is relatively easy to determine
(2) Whenever gap > gap2
the growth of gap slows down
while gap2 grows constantly except at Ramsey
If s (t+1)  s2 (t+1), then s (t+1)  s (t)+3
s2 (t+1)=s2 (t) + 2, unless s2 (t)+1 or +2 is Ramsey

14. The gap of triangle-free graphs
The more you want, the less you get :
For gap-critical graphs,(if they are connected):
 = n / 2 , so gap = n / 2 - (n)
If not connected, apply to components :
the components are connected, gap-critical

15. If G is gap-critical, (G) > (G-v) for all v.
If G is triangle-free, connected and
(G) > (G-v) for all v, then (G) = n / 2
Gallai : For triangle-free :  +  =n
Gallai : If  (G-v) =  (G) for every vertex v,
then G-v has a perfect matching –’’--.
In particular, n is odd

16. Triangle-free, connected cont’d :
THEN GAP-CRITICAL FACTOR-CRITICAL
Proof of Gallai’s Lemma : …
Equivalent to Tutte’s theorem ( Tutte-set)

17. The gap of triangle free graphs
Theorem : gap2 (n) = n / 2 - (n) , or +1
the latter  n is even Ramsey-perfect.
Proof: At most 2 comp: from inequalities about Ramsey:
if n =n1 + n2 + n3 , then (n) < (n1) + (n2) + (n3)
2 comp : if of only if n is even, Ramsey- perfect, and the
components have odd size n1, n2 , (n) = (n1) + (n2)
For the connected components by Gallai:  = n / 2

18. The proof relies on two lucky facts :
(1) gap2 is relatively easy to determine
(2) Whenever gap > gap2
the growth of gap slows down
while gap2 grows constantly except at Ramsey
If s (t+1) < s2 (t+1), then s (t+1)  s (t)+3
n / 2 - (n)

19. Theorem: 0  gap(n) - gap2 (n)  2 and they are equal everywhere but on small constant size intervals after Ramsey numbers.
Corollary: All subgraphs of (3,  ) Ramsey-graphs
of order at least R(3, -1) are perfectly matchable. T
s(t) or 21, 32 or
R , 9, , 18, ,28,

20. 0  s2(t) - s (t)  10 n / 2-(n)  gap(n)  n / 2 - (n)+3
= almost always
NOT YET THE END
The conjectures ?
Dramatic corollaries for Ramsey ?
s(5) = 20 or 21 (= s2(5) ) ?