Polynomial Chi-binding functions and forbidden induced subgraphs a - PowerPoint PPT Presentation

Polynomial Chi-binding functions and forbidden induced subgraphs – a survey Ingo Schiermeyer TU Bergakademie Freiberg Chromatic Number Ingo Schiermeyer GGTW 2019

Chromatic Number András Gyárfás (Budapest, 1985) Problems from the world surrounding perfect graphs Chromatic Number Ingo Schiermeyer GGTW 2019

Chromatic Number Conjecture (Gyárfás) f Let T be any tree (or forest). Then there is a function T  ( -binding function) such that every T-free graph G satisfies   T  (G) f ( ( G)) Chromatic Number Ingo Schiermeyer GGTW 2019

Chromatic Number Theorem (Erdös, 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 GGTW 2019

Chromatic Number Mycielski construction  =  = A sequence of graphs G , G ,... with (G ) 2 and (G ) k. 2 3 k k = = G K , G C , G is the Grötzsch/M ycielski graph 2 2 3 5 4 Chromatic Number Ingo Schiermeyer GGTW 2019

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 GGTW 2019

Chromatic Number Theorem (Gyárfás, 1987) If T is the path P then k    + R( k/2 , 1) - 1     (G) - 1 (G) (k - 1)   k/2 - 1 Theorem (Esperet, Lemoine, Maffray, Morel, 2013) If T is the path P then 5     (G) - 3 (G) 5 3 Chromatic Number Ingo Schiermeyer GGTW 2019

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. The SPGC became the Strong Perfect Graph Theorem in 2002 by Chudnovsky, Robertson, Seymour, and Thomas. Chromatic Number Ingo Schiermeyer GGTW 2019

Chromatic Number Problem (Gyárfás, 1987) What is the order of magnitude of f ? P 5 Theorem (Gyárfás,1987) There is no linear binding function for P5-free graphs.   Known : Let H be a graph with (H) 3, then  2 (G)   f(P , H) c  5 log Chromatic Number Ingo Schiermeyer GGTW 2019

Chromatic Number This paper of (Gyárfás) contains several other challenging conjectures. Polynomial chi-Binding Functions and Forbidden Induced Subgraphs: A Survey Bert Randerath and Ingo Schiermeyer Graphs and Combinatorics 2019 Chromatic Number Ingo Schiermeyer GGTW 2019

Chromatic Number Quadratic B Bounds Linear Bounds Vizing Bound Perfect Graphs Chromatic Number Ingo Schiermeyer GGTW 2019

Induced subgraphs Banner Diamond House Bull Gem Chromatic Number Ingo Schiermeyer GGTW 2019

Chromatic Number Theorem (Fouquet et al. 1995)       2 f(P , House)   5   Theorem (IS 2014) Parachute   2 f(P , H) 5  for H {claw, paw, diamond, dart, gem, cricket, parachute} Chromatic Number Ingo Schiermeyer GGTW 2019

Induced subgraphs Banner Theorem (Brause, Geisser, Randerath, and Schiermeyer 2019+)  2 ( G ) •    2 c f(P , banner) ( G )  5 log Chromatic Number Ingo Schiermeyer GGTW 2019

Chromatic Number Theorem   + f(P , diamond) (G) 1 5 diamond Theorem (Cameron, Huang, and Merkel 2018+)   + f(P , diamond) (G) 3 6 Chromatic Number Ingo Schiermeyer GGTW 2019

Chromatic Number Theorem (Chudnovsky et al. 2019+)    5 (G)  f(P , gem)   5   4 gem Theorem (Randerath and Schiermeyer 2019)    f(P , gem) (k - 2)( (G) - 1) for k 4 k Chromatic Number Ingo Schiermeyer GGTW 2019

Chromatic Number Theorem (Chudnovsky and Sivaraman 2019)  +   ( G ) 1    f(P , bull)   5   2   (G) - 1 f(P , C ) 2 5 5 Problem: Can the chi-binding function for (P5,C5)- free graphs be improved? Why is C5 so difficult? Chromatic Number Ingo Schiermeyer GGTW 2019

Induced subgraphs Butterfly -- Windmill (3,2) Chromatic Number Ingo Schiermeyer GGTW 2019

Chromatic Number Windmill graphs Chromatic Number Ingo Schiermeyer GGTW 2019

Chromatic Number Theorem (IS 2015)  p Let H be the windmill W for some p 2. Then 3   2p - 1 f(P , H) . 5 Chromatic Number Ingo Schiermeyer GGTW 2019

Chromatic Number Theorem (Brause, Doan, and Schiermeyer 2015)  Let G be a (P , K ) - free graph for some t 2. 5 2, t    t Then (G) c for a constant c . t t Chromatic Number Ingo Schiermeyer GGTW 2019

Induced subgraphs Parachute Twin-Parachute Chromatic Number Ingo Schiermeyer GGTW 2019

Chromatic Number Problem Does there exist a polynomial chi - binding function for (P , Twin - parachute) - free graphs? 5 Chromatic Number Ingo Schiermeyer GGTW 2019

Chromatic Number Landscape 2016 P5 ? ? ? ? ? P5, gem+ P5, cricket P5, dart P5, gem P5,hammer P5, windmill P5,claw P5, paw P5, house P5, K2,t P5, diamond Chromatic Number Ingo Schiermeyer GGTW 2019

Chromatic Number P5,C5 Landscape 2019 P5 ? ? ? P5,paraglider P5,kite ? ? P5, parachute P5,banner P5,bull P5, cricket P5, dart P5, gem P5,hammer P5, windmill P5,claw P5, paw P5, house P5, K2,t P5, diamond Chromatic Number Ingo Schiermeyer GGTW 2019

Chromatic Number Chromatic Number Ingo Schiermeyer GGTW 2019

Chromatic Number Chromatic Number Ingo Schiermeyer GGTW 2019

Chromatic Number Dōmo arigatō gozaimasu! Herzlichen Dank! Thank you very much! Chromatic Number Ingo Schiermeyer GGTW 2019

The end Chromatic Number Ingo Schiermeyer GGTW 2019