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On polynomial chi-binding functions
Ingo Schiermeyer
TU Bergakademie Freiberg
BIRS 2016
Chromatic Number
Ingo Schiermeyer
Chromatic Number
Theorem (Erdős, 1959)
For every k, g  3, there is a graph G with girth g and
chromatic number k.
Coloring, sparseness, and girth, 2015
Alon, Kostochka, Reiniger, West, and Zhu
Chromatic Number
Ingo Schiermeyer
Chromatic Number
Mycielski construction
A sequence of graphs G 2 , G 3 ,... with  (G k )  2 and  (G k )  k.
G 2  K 2 , G 3  C5 , G 4 is the Grötzsch/M ycielski graph
Chromatic Number
Ingo Schiermeyer
Chromatic Number
Strong Perfect Graph Conjecture (Berge, 1960)
A graph G is perfect if and only if it does not contain an odd
hole nor an odd antihole of length at least five.
The SPGC became the
Strong Perfect Graph Theorem in 2002 by Chudnovsky,
Robertson, Seymour, and Thomas.
Chromatic Number
Ingo Schiermeyer
Chromatic Number
András Gyárfás (Budapest, 1985)
Problems from the world surrounding
perfect graphs
Chromatic Number
Ingo Schiermeyer
Chromatic Number
Conjecture (Gyárfás)
Let T be any tree (or forest). Then there is a function f T
( -binding function) such that every T-free graph G satisfies
 (G)  f T ((G))
Chromatic Number
Ingo Schiermeyer
Chromatic Number
This paper of (Gyárfás) contains several other challenging
conjectures.
Project „Induced subgraphs of graphs with large chromatic
number“ by Chudnovsky, Scott, and Seymour.
Theorem (CSS 2014)
Every pentagonal graph is 82200-colourable.
Chromatic Number
Ingo Schiermeyer
Chromatic Number
Theorem (Gyárfás, 1987)
There is a chi-binding function for
 Stars
 Paths
 Brooms
 Trees with radius 2 (Kierstead and Penrice)
 Special trees with radius 3 (Kierstead and Zhu)
 pK2 (Wagon 1980)
Chromatic Number
Ingo Schiermeyer
Chromatic Number
Problem 2.7 (Gyárfás, 1987)
What is the order of magnitude of f P5 ?
Problem 2.16 (Gyárfás, 1987)
What is the order of magnitude of f 2K 2 ?
Chromatic Number
Ingo Schiermeyer
Chromatic Number
Problem 2.7 (Gyárfás, 1987)
What is the order of magnitude of f P5 ?
If there is a polynomial binding function for P5-free graphs,
then it would imply the
Erdös-Hajnal Conjecture for P5-free graphs, which is open.
Theorem (Gyárfás,1987)
There is no linear binding function for P5-free graphs.
Chromatic Number
Ingo Schiermeyer
Chromatic Number
Erdös-Hajnal Conjecture
For every fixed subgraph H there exists an  (H) such that
if G is an H - free graph of order n then it has a clique or
a stable set of size n  (H) .
Chromatic Number
Ingo Schiermeyer
Induced subgraphs
Dart
Gem
House
Bull
Kite
Chromatic Number
Ingo Schiermeyer
Induced subgraphs
House
K_2,3
Chromatic Number
Ingo Schiermeyer
Chromatic Number
Problem 2.16 (Gyárfás, 1987)
What is the order of magnitude of f 2K 2 ?
Theorem (Wagon, 1980)
   1
.
If G is 2K 2 -free, then  (G)  
 2 
Theorem (Chung, 1980)
There is no linear binding function.
Theorem (BRSV, 2016)
If G is (2K 2 , claw) - free, then G has
no linear binding function.
Chromatic Number
Ingo Schiermeyer
Chromatic Number
Problem 2.16 (Gyárfás, 1987)
What is the order of magnitude of f 2K 2 ?
Theorem (Wagon, 1980)
   1
.
If G is 2K 2 -free, then  (G)  
 2 
f * (2)  3, f * (3)  4, 6  f * (4)  10
Work in progress
Chromatic Number
Ingo Schiermeyer
Chromatic Number
C4
2K 2   1
P5
 5 
 4 
K 4  e Gem
P5
 1
??
   1


2


K1,3 ,   3
??
??
??
6
 5 
5 
 4 
??
8
P6
P3  P2
W4
 3
Chromatic Number
Ingo Schiermeyer
Chromatic Number
C4
2K 2   1
P5
 5 
 4 
K 4  e Gem
P5
3

2
 1
   1


2


K1,3 ,   3
??
??
??
6
 5 
5 
 4 
??
8
P6
P3  P2
W4
 3
Chromatic Number
Ingo Schiermeyer
Chromatic Number
C4
2K 2   1
P5
 5 
 4 
K 4  e Gem
P5
3

2
 1
   1


2


K1,3 ,   3
2
??
??
6
 5 
5 
 4 
??
8
P6
P3  P2
W4
 3
Chromatic Number
Ingo Schiermeyer
Chromatic Number
C4
2K 2   1
P5
 5 
 4 
K 4  e Gem
P5
3

2
 1
   1


2


K1,3 ,   3
2
5  5
??
6
 5 
5 
 4 
??
8
P6
P3  P2
W4
 3
Chromatic Number
Ingo Schiermeyer
Chromatic Number
C4
2K 2   1
P5
 5 
 4 
K 4  e Gem
P5
3

2
 1
   1


2


K1,3 ,   3
2
5  5 perfect
6
 5 
5 
 4 
??
8
P6
P3  P2
W4
 3
Chromatic Number
Ingo Schiermeyer
Chromatic Number
C4
2K 2   1
P5
 5 
 4 
K 4  e Gem
P5
3

2
 1
   1


2


K1,3 ,   3
2
5  5 perfect
6
 5 
5 
 4 
perfect
8
P6
P3  P2
W4
 3
Chromatic Number
Ingo Schiermeyer
Chromatic Number
Theorem (Fouquet et al. 1995)
 
f(P5 , House)   
2 
Theorem (IS 2014)
Gem+
f(P5 , H)   2
for H {Claw, Paw, Diamond, Dart, Gem, Cricket, Gem  }
Chromatic Number
Ingo Schiermeyer
Induced subgraphs
Gem+
TwinGem
Chromatic Number
Ingo Schiermeyer
Chromatic Number
Problem
Does there exist a polynomial chi - bounding function
for (P5 , Twingem) - free graphs?
Chromatic Number
Ingo Schiermeyer
Chromatic Number
P5
P5, gem+
? ? ?
? ?
P5, cricket
P5, dart
P5, gem
P5,claw
P5, paw
P5, diamond
Chromatic Number
P5,hammer
P5, house
P5, windmill
P5, K2,t
Ingo Schiermeyer
Chromatic Number
Theorem (Gyárfás, 1987)
If T is the path Pk then
R( k/2 ,   1) - 1
  (G)  (k - 1) (G)-1
k/2  - 1
Theorem (Esperet, Lemoine, Maffray, Morel, 2013)
If T is the path P5 then
 (G)  5  3 (G)-3
f * (2)  3, f * (3)  5, 8  f * (4)  15
Chromatic Number
Ingo Schiermeyer
Chromatic Number
Theorem (Gyárfás, 1987)
If T is the star Sk  K1,k -1 then
R(k - 1,   1) - 1
  (G)  R(k - 1,  )
k-2
k  3 : f * ( x)  x
(perfect graphs)
k  4 : f * (2)  3, 4  f * (3)  5
(claw - free graphs)
k  5 : f * (2)  3, 6  f * (3)  8
Chromatic Number
Ingo Schiermeyer
Chromatic Number
Dōmo arigatō gozaimasu!
Herzlichen Dank!
Mille grazie!
Thank you very much!
Chromatic Number
Ingo Schiermeyer
The end
Chromatic Number
Ingo Schiermeyer
Chromatic Number
Theorem (BDS 2015)
Let G be a (P5 , K 2,t ) - free graph for some t  2.
Then  (G)  c t  t for a constant c t .
Chromatic Number
Ingo Schiermeyer
Chromatic Number
Sketch of proof:
 By the Strong Perfect Graph Theorem every non perfect
(P5 , K 2,t ) - free graph contains an induced C5 or an
induced C 2p1 for some 3    p.
 We consider t he neighbourh ood of the C5 .
 Using the Ramsey number R(K  , K t ) we manage to find
an induced K 2,t or to show that  (G)  c t  t .
Chromatic Number
Ingo Schiermeyer
Induced subgraphs
Butterfly -- Windmill (3,2)
CTW 2016
Chromatic Number
Ingo Schiermeyer
Chromatic Number
CTW 2016
Chromatic Number
Ingo Schiermeyer
SEG 2016
Chromatic Number
Ingo Schiermeyer
Chromatic Number
Problems:
Can 4 - colourabil ity be solved in polynomial time
for the class of
 (P6 , bull) - free graphs?
 (P6 , Z2) - free graphs?
 (P6 , kite) - free graphs?
 (P6 , dart) - free graphs?
 (P6 , cricket) - free graphs?
We are still confused, but on a higher level (Feynmann)
Chromatic Number
Ingo Schiermeyer
Induced subgraphs
Dart
Gem
House
Bull
Kite
Chromatic Number
Ingo Schiermeyer
Chromatic Number
Theorem (Fouquet et al. 1995)
 
f(P5 , House)   
2 
Theorem (IS 2014)
Gem+
f(P5 , H)   2
for H {Claw, Paw, Diamond, Dart, Gem, Cricket, Gem  }
Chromatic Number
Ingo Schiermeyer
Induced subgraphs
Gem+
TwinGem
Chromatic Number
Ingo Schiermeyer
Chromatic Number
Problem
Does there exist a polynomial chi - bounding function
for (P5 , Twingem)?
Chromatic Number
Ingo Schiermeyer
Chromatic Number
András Gyárfás (Budapest, 1987)
Problems from the world surrounding
perfect graphs
AGTAC 2015
Chromatic Number
Ingo Schiermeyer
Chromatic Number
Conjecture (Gyárfás)
Let T be any tree (or forest). Then there is a function f T
( -binding function) such that every T-free graph G satisfies
 (G)  f T ((G))
SEG 2015
Chromatic Number
Ingo Schiermeyer
Chromatic Number
Mycielski construction
A sequence of graphs G 2 , G 3 ,... with  (G k )  2 and  (G k )  k.
G 2  K 2 , G 3  C5 , G 4 is the Grötzsch/M ycielski graph
SEG 2015
Chromatic Number
Ingo Schiermeyer
Chromatic Number
Theorem (Gyárfás, 1987)
If T is a path Pk with k  3 then
 (G)  (k - 1) (G)-1
Theorem (Esperet, Lemoine, Maffray, Morel, 2013)
If T is the path P5 then
 (G)  5  3 (G)-3
SEG 2015
Chromatic Number
Ingo Schiermeyer
Chromatic Number
Problem (Gyárfás,1987)
What is the order of magnitude of f P5 ?
If there is a polynomial binding function for P5-free graphs,
then it would imply the
Erdös-Hajnal Conjecture for P5-free graphs, which is open.
Theorem (Gyárfás,1987, IS 2014)
There is no linear binding function for P5-free graphs.
SEG 2015
Chromatic Number
Ingo Schiermeyer
Chromatic Number
Theorem (Fouquet et al. 1995)
 
f(P5 , House)   
2 
Theorem (IS 2014)
f(P5 , H)   2
for H {Claw, Paw, Diamond, Dart, Gem, Cricket, Gem  }
SEG 2015
Chromatic Number
Ingo Schiermeyer
Chromatic Number
Theorem (Brooks, 1941)
Every connected graph G satisfies  (G)  (G) unless G is
complete or an odd cycle.
SEG 2015
Chromatic Number
Ingo Schiermeyer
Chromatic Number
Conjecture (Borodin and Kostochka, 1977)
Every graph G with maximum degree   9 has chromatic number
 (G)  max { - 1, }.
Proved for   1014 by Reed 1999.
Proved for claw-free graphs by Cranston and Rabern 2012.
Theorem (Rabern, 2014)
Every graph G satisfies  (G)  max { - 1,  ,4 }.
SEG 2015
Chromatic Number
Ingo Schiermeyer
Chromatic Number
Conjecture (Reed, 1998)
  (G)  (G)  1
Every graph G satisfies  (G)  
.

2


SEG 2015
Chromatic Number
Ingo Schiermeyer
Chromatic Number
Reed‘s conjecture holds for
 graphs in which t he complement is disconnect ed
n  3 -
2
 graphs satisfying   n  2 - (  n  5 -  )
 graphs satisfying  
8(n -  )  118
21
 Line graphs (King, Reed, Vetta, 2006)
 K 3 - free graphs with  
SEG 2015
Chromatic Number
Ingo Schiermeyer
Chromatic Number
Reed‘s conjecture holds for
 (P5 , C 4 ) - free graphs
 (P5 , House, Dart) - free graphs
 (P5 , Kite, Bull) - free graphs
Fouquet, Vanherpe, 2011
SEG 2015
Chromatic Number
Ingo Schiermeyer
Induced subgraphs
House
Bull
SEG 2015
Dart
Gem
Kite
Chromatic Number
Ingo Schiermeyer
Chromatic Number
Theorem (IS 2014)
For every fixed k  5 and   3 there exists n( ) such that
if G is a connected Pk - free graph of order n  n( ), then
     1
 (G)  
.

2


For k  5 we need n  100  32 (G)-6
SEG 2015
Chromatic Number
Ingo Schiermeyer
Chromatic Number
Theorem (IS 2014)
Let G be a connected (P5 , H) - free graph of order n and
     1
clique number  with 2    n1/3. Then   
for

2


H {Claw, Paw, Diamond, House, Dart, Gem, Cricket, Gem  }.
SEG 2015
Chromatic Number
Ingo Schiermeyer
Chromatic Number
Ingo Schiermeyer
Chromatic Number
Erdös, P., Simonovits, M., Sos, V.T.: Anti-Ramsey theorems.
In: Infinite and Finite Sets (Proc.Colloq. Keszthely, Hungary 1973) edited by
Hajnal, A., Rado, R., Sos, V.T. vol. II, pp. 657-665.
Colloq. Math. Soc. Janos Bolyai 10. Amsterdam: North-Holland 1975
MCW XX
Chromatic Number
Ingo Schiermeyer
Chromatic Number
f(G,H): Maximum number of colours in an
edge colouring of G with no totally
multicoloured copy of H.
If G is coloured with at least f(G,H) + 1 =
rb(G,H) colours, then there is a totally
multicoloured copy or a Rainbow copy of H.
MCW XX
Chromatic Number
Ingo Schiermeyer
Chromatic Number
Let H  {H - e | e  E(H)} . Then
ext(G, H )  1  f(G, H) ,
ext(G, H )  2  f(G, H)  1  rb(G, H).
MCW XX
Chromatic Number
Ingo Schiermeyer
Chromatic Number
Turan numbers
Extremal graph theory
Rainbow numbers
MCW XX
Chromatic Number
Ingo Schiermeyer
Chromatic Number
The Turán graph Tnk has
k -1 2 2  i 
t 
(n - i )    edges, where n  tk  i with
2k
 2
k
n
0  i  k - 1 and k  n.
Theorem (Turán, 1941)
Every graph G of order n and with m  t kn  ext(K n , K k 1 )
edges contains a complete subgraph K k 1.
MCW XX
Chromatic Number
Ingo Schiermeyer
Chromatic Number
Theorem (Erdös, Simonovits and Sós, 1975)
rb(K n , K 3 )  n for all n  3.
Theorem (Montellano-Ballesteros and Neumann-Lara,
Schiermeyer, 2002)
rb(K n , K k )  ext(K n , K k-1 )  2 for all n  k  4.
MCW XX
Chromatic Number
Ingo Schiermeyer
Rainbow Matchings
Theorem (Erdös & Gallai, 1959)
 2k - 1  k - 1
, 
  (k - 1)(n - k  1)}  ext(K n , kK 2 ),
If m  max{ 
 2  2 
then kK 2  G.
Theorem (Schiermeyer, 2004)
rb(K n , kK2 )  ext(K n , (k -1)K 2 )  2 for all k  2 and n  3k  3.
MCW XX
Chromatic Number
Ingo Schiermeyer
Rainbow Matchings
K 2k-3
MCW XX
K k -2
Chromatic Number
Ingo Schiermeyer
Rainbow Cycles
Conjecture (Erdös, Simonovits and Sós, 1975)
f(K n , C k )  n(
k-2
1
k -1 - r
1

) - r(

)
2
k -1
2
k -1
with n = t(k-1) + r and 1  r  k - 1
This conjecture was proved for k=4 by Alon (1983), for k = 5, 6 and 7
by Jiang, West and Schiermeyer (2001, 2003), and for all k by
Montellano-Ballesteros and Neumann-Lara (2005).
MCW XX
Chromatic Number
Ingo Schiermeyer
Rainbow Cycles
f(K n , C k )  n(
MCW XX
k-2
1
k -1 - r
1

) - r(

)
2
k -1
2
k -1
Chromatic Number
Ingo Schiermeyer
Rainbow Cycles
Theorem (Montellano-Ballesteros, Neumann-Lara, 2005)
 n  k - 1  r   n 

     
rb(K n , C k )  


 k - 1 2   2   k - 1
with n = t(k-1) + r and 0  r  k - 2
MCW XX
Chromatic Number
Ingo Schiermeyer
Rainbow Cycles
Theorem (Gorgol, 2008)
rb(K n , Ck )  rb(K n , Ck ), n  k  1
where Ck denotes a cycle Ck with a pendant edge.
Theorem (Gorgol, 2008)
rb(K n , Ck  )  rb(K n , Ck ), n  k  2
where C k  denotes a cycle Ck with two pendant edges.
MCW XX
Chromatic Number
Ingo Schiermeyer
Rainbow bulls and diamonds
K 1,3  e
K1,4  e
Bull B
MCW XX
Diamond D
Chromatic Number
Ingo Schiermeyer
Rainbow numbers for
certain graphs
Theorem (Gorgol and Łazuka, 2006)
rb(K n , K1,3  e)  n for n  4.
rb(K n , K1,4  e)  n  2 for n  5.
Theorem (Neupauerova, Sotak, IS 2012)
rb(K 5 , B)  6,
rb(K n , B)  n  2 for n  6.
MCW XX
Chromatic Number
Ingo Schiermeyer
Rainbow numbers for diamonds
Theorem (IS, 2012)
rb(K n , D)   (n 3/2 )
Theorem (IS, 2012)
Let H be a graph of order p  k  3 containing a cycle Ck .
If H has cyclomatic number v ( H )  2, then rb(K n , H)
has no upper bound which is linear in n.
MCW XX
Chromatic Number
Ingo Schiermeyer
Rainbow numbers for diamonds
Theorem (Garnick, Kwong, and Lazebnik, 1993)
1/(2 2 )  lim inf ext(n, C3 , C 4 )/n 3/2  lim sup ext(n, C3 , C 4 )/n 3/2  1/2
n 
n 
Conjecture (Erdös, 1975)
ext(n, C3 , C 4 )  (
1
2 2
 o(1)) n 3/2
Theorem (Garnick, Kwong, and Lazebnik, 1993)
1
ext(n, C3 , C 4 )  n n - 1
2
MCW XX
Chromatic Number
Ingo Schiermeyer
Rainbow numbers for diamonds
Theorem (IS, 2012)
ext(K 4 , C3 , C 4 )  3, rb(K 4 , D)  5
ext(K 5 , C3 , C 4 )  5,
rb(K 5 , D)  7
ext(K 6 , C3 , C 4 )  6,
rb(K 6 , D)  8
ext(K 7 , C3 , C 4 )  8,
rb(K 7 , D)  10
ext(K 8 , C3 , C 4 )  10,
rb(K 8 , D)  12
ext(K 9 , C3 , C 4 )  12,
rb(K 9 , D)  14
ext(K 11 , C3 , C 4 )  16
MCW XX
Chromatic Number
Ingo Schiermeyer
Rainbow numbers for diamonds
Theorem (IS, 2012)
ext(K 4 , C3 , C 4 )  3,
rb(K 4 , D)  5
ext(K 5 , C3 , C 4 )  5,
rb(K 5 , D)  7
ext(K 6 , C3 , C 4 )  6,
rb(K 6 , D)  8
ext(K 7 , C3 , C 4 )  8,
rb(K 7 , D)  10
ext(K 8 , C3 , C 4 )  10,
rb(K 8 , D)  12
ext(K 9 , C3 , C 4 )  12,
rb(K 9 , D)  14
ext(K 10 , C3 , C4 )  15, rb(K10 , D)  17
ext(K 11, C3 , C4 )  16
MCW XX
Chromatic Number
Ingo Schiermeyer
Rainbow numbers for diamonds
rb(K10 , D)  17
MCW XX
Chromatic Number
Ingo Schiermeyer
Rainbow numbers for diamonds
177  rb(K 50 , D)  226
MCW XX
Chromatic Number
Ingo Schiermeyer
Rainbow Colourings
MCW XX
n
D
H
K(2,3)
4
5
5
7
8
8
6
8
9
10
7
10
11
12
8
12
14
9
14
10
17
Chromatic Number
Ingo Schiermeyer
GRAPHS 2008
Chromatic Number
Ingo Schiermeyer