On polynomial chi-binding functions Ingo Schiermeyer TU Bergakademie Freiberg BIRS 2016 1

Chromatic Number Theorem (Erdős, 1959) Coloring, sparseness, and girth, 2015 Alon, Kostochka, Reiniger, West, and Zhu

Chromatic Number Mycielski construction

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 András Gyárfás (Budapest, 1985) Problems from the world surrounding perfect graphs

Chromatic Number Conjecture (Gyárfás) Let T be any tree (or forest). Then there is a function ( -binding function) such that every T-free graph G satisfies

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 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 Problem 2.7 (Gyárfás, 1987)

Chromatic Number Problem 2.7 (Gyárfás, 1987) 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. Problem 2.7 (Gyárfás, 1987) Theorem (Gyárfás,1987) There is no linear binding function for P5-free graphs.

Chromatic Number Erdös-Hajnal Conjecture

Induced subgraphs Dart Gem House Bull Kite

Induced subgraphs K_2,3 House

Chromatic Number Problem 2.16 (Gyárfás, 1987) Theorem (Wagon, 1980) Theorem (Chung, 1980) There is no linear binding function. Theorem (BRSV, 2016)

Chromatic Number Problem 2.16 (Gyárfás, 1987) Theorem (Wagon, 1980) Work in progress

Chromatic Number ? ?

Chromatic Number ? ?

Chromatic Number ? ?

Chromatic Number ? ?

Chromatic Number ? ?

Chromatic Number

Chromatic Number Theorem (Fouquet et al. 1995) Theorem (IS 2014) Gem+

Induced subgraphs Gem+ TwinGem

Chromatic Number Problem

Chromatic Number P5 ? ? ? ? ? P5, gem+ P5, cricket P5, dart P5, gem ? ? ? ? ? P5, gem+ P5, cricket P5, dart P5, gem P5,hammer P5, windmill P5,claw P5, paw P5, diamond P5, house P5, K2,t

Chromatic Number Theorem (Gyárfás, 1987) Theorem (Esperet, Lemoine, Maffray, Morel, 2013)

Chromatic Number Theorem (Gyárfás, 1987)

Chromatic Number Dōmo arigatō gozaimasu! Herzlichen Dank! Mille grazie! Thank you very much!

The end

Chromatic Number Theorem (BDS 2015)

Chromatic Number Sketch of proof:

Induced subgraphs Butterfly -- Windmill (3,2) CTW 2016

Chromatic Number CTW 2016

SEG 2016

Chromatic Number Problems: We are still confused, but on a higher level (Feynmann)

Induced subgraphs Dart Gem House Bull Kite

Chromatic Number Theorem (Fouquet et al. 1995) Theorem (IS 2014) Gem+

Induced subgraphs Gem+ TwinGem

Chromatic Number Problem

Chromatic Number András Gyárfás (Budapest, 1987) Problems from the world surrounding perfect graphs AGTAC 2015

Chromatic Number Conjecture (Gyárfás) Let T be any tree (or forest). Then there is a function ( -binding function) such that every T-free graph G satisfies SEG 2015

Chromatic Number Mycielski construction SEG 2015

Chromatic Number Theorem (Gyárfás, 1987) Theorem (Esperet, Lemoine, Maffray, Morel, 2013) SEG 2015

Chromatic Number Problem (Gyárfás,1987) 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 Theorem (Fouquet et al. 1995) Theorem (IS 2014) SEG 2015

Chromatic Number Theorem (Brooks, 1941) SEG 2015

Chromatic Number Conjecture (Borodin and Kostochka, 1977) Proved for claw-free graphs by Cranston and Rabern 2012. Theorem (Rabern, 2014) SEG 2015

Chromatic Number Conjecture (Reed, 1998) SEG 2015

Chromatic Number Reed‘s conjecture holds for SEG 2015

Chromatic Number Reed‘s conjecture holds for SEG 2015

Induced subgraphs Dart Gem House Bull Kite SEG 2015

Chromatic Number Theorem (IS 2014) SEG 2015

Chromatic Number Theorem (IS 2014) SEG 2015

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 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 Let . Then MCW XX

Chromatic Number Turan numbers Extremal graph theory Rainbow numbers MCW XX

Chromatic Number Theorem (Turán, 1941) MCW XX

Chromatic Number Theorem (Erdös, Simonovits and Sós, 1975) Theorem (Montellano-Ballesteros and Neumann-Lara, Schiermeyer, 2002) MCW XX

Rainbow Matchings Theorem (Erdös & Gallai, 1959) Theorem (Schiermeyer, 2004) MCW XX

Rainbow Matchings MCW XX

Rainbow Cycles Conjecture (Erdös, Simonovits and Sós, 1975) with n = t(k-1) + r and 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

Rainbow Cycles MCW XX

Rainbow Cycles Theorem (Montellano-Ballesteros, Neumann-Lara, 2005) with n = t(k-1) + r and MCW XX

Rainbow Cycles Theorem (Gorgol, 2008) where denotes a cycle with a pendant edge. Theorem (Gorgol, 2008) where denotes a cycle with two pendant edges. MCW XX

Rainbow bulls and diamonds Bull B Diamond D MCW XX

Rainbow numbers for certain graphs Theorem (Gorgol and Łazuka, 2006) Theorem (Neupauerova, Sotak, IS 2012) MCW XX

Rainbow numbers for diamonds Theorem (IS, 2012) Theorem (IS, 2012) Let H be a graph of order containing a cycle If H has cyclomatic number , then has no upper bound which is linear in n. MCW XX

Rainbow numbers for diamonds Theorem (Garnick, Kwong, and Lazebnik, 1993) Conjecture (Erdös, 1975) Theorem (Garnick, Kwong, and Lazebnik, 1993) MCW XX

Rainbow numbers for diamonds Theorem (IS, 2012) MCW XX

Rainbow numbers for diamonds Theorem (IS, 2012) MCW XX

Rainbow numbers for diamonds MCW XX

Rainbow numbers for diamonds MCW XX

Rainbow Colourings n D H K(2,3) 4 5 7 8 6 9 10 11 12 14 17 MCW XX

GRAPHS 2008