The independence numbers and the chromatic numbers of random - PowerPoint PPT Presentation

The independence numbers and the chromatic numbers of random subgraphs of Kneser’s graphs and their generalizations Andrei Raigorodskii Moscow Institute of Physics and Technology, Yandex, Moscow, Russia Shanghai, China, 26.03.2017 A. Raigorodskii (MIPT, YND) 2017 Shanghai 1 / 11

� [ n ] Let [ n ] = { 1 ; 2 ; : : : ; n } . Assume that F ⊂ r � n= 2 is such a collection r n − 1 r -subsets that any two of them intersect. Then |F| � r − 1 The Erd˝ os–Ko–Rado Theorem Erd˝ os–Ko–Rado, 1961 � with � � of . A. Raigorodskii (MIPT, YND) 2017 Shanghai 2 / 11

� [ n ] Let [ n ] = { 1 ; 2 ; : : : ; n } . Assume that F ⊂ r � n= 2 is such a collection r n − 1 r -subsets that any two of them intersect. Then |F| � r − 1 n − 1 r − 1 The Erd˝ os–Ko–Rado Theorem Erd˝ os–Ko–Rado, 1961 � with � � of . � � Of course the bound is attained on a “star”. A. Raigorodskii (MIPT, YND) 2017 Shanghai 2 / 11

� [ n ] Let [ n ] = { 1 ; 2 ; : : : ; n } . Assume that F ⊂ r � n= 2 is such a collection r n − 1 r -subsets that any two of them intersect. Then |F| � r − 1 n − 1 r − 1 The Erd˝ os–Ko–Rado Theorem � [ n ] Let [ n ] = { 1 ; 2 ; : : : ; n } . Assume that F ⊂ r � n= 2 is such a collection r Erd˝ os–Ko–Rado, 1961 r -subsets that any two of them intersect and F is not a star. Then n − 1 n − r − 1 � with r − 1 r − 1 � � of . � � Of course the bound is attained on a “star”. Hilton–Milner, 1967 � with of � � � � |F| � − + 1. A. Raigorodskii (MIPT, YND) 2017 Shanghai 2 / 11

� [ n ] Let [ n ] = { 1 ; 2 ; : : : ; n } . Assume that F ⊂ r � n= 2 is such a collection r n − 1 r -subsets that any two of them intersect. Then |F| � r − 1 n − 1 r − 1 The Erd˝ os–Ko–Rado Theorem � [ n ] Let [ n ] = { 1 ; 2 ; : : : ; n } . Assume that F ⊂ r � n= 2 is such a collection r Erd˝ os–Ko–Rado, 1961 r -subsets that any two of them intersect and F is not a star. Then n − 1 n − r − 1 � with r − 1 r − 1 � � of . � � Of course the bound is attained on a “star”. Hilton–Milner, 1967 � with of � � � � |F| � − + 1. It’s a famous stability result. A. Raigorodskii (MIPT, YND) 2017 Shanghai 2 / 11

� [ n ] Let [ n ] = { 1 ; 2 ; : : : ; n } . Assume that F ⊂ r � n= 2 is such a collection r n − 1 r -subsets that any two of them intersect. Then |F| � r − 1 n − 1 r − 1 The Erd˝ os–Ko–Rado Theorem � [ n ] Let [ n ] = { 1 ; 2 ; : : : ; n } . Assume that F ⊂ r � n= 2 is such a collection r Erd˝ os–Ko–Rado, 1961 r -subsets that any two of them intersect and F is not a star. Then n − 1 n − r − 1 � with r − 1 r − 1 � � of . � � Of course the bound is attained on a “star”. Hilton–Milner, 1967 � with of � � � � |F| � − + 1. It’s a famous stability result. Other stability results were proposed by Balogh, Bohman, Mubayi et al. using the notion of a random hypergraph. A. Raigorodskii (MIPT, YND) 2017 Shanghai 2 / 11

A graph-theoretic point of view A. Raigorodskii (MIPT, YND) 2017 Shanghai 3 / 11

� ( G ) of a graph G is the maximum number of G . A graph-theoretic point of view The independence number pairwise disjoint vertices of A. Raigorodskii (MIPT, YND) 2017 Shanghai 3 / 11

� ( G ) of a graph G is the maximum number of G . � [ n ] K G V ; E ), where V = n;r = ( r E = { ( A; B ) : A ∩ B = ∅} : A graph-theoretic point of view The independence number pairwise disjoint vertices of Kneser’s graph � , A. Raigorodskii (MIPT, YND) 2017 Shanghai 3 / 11

� ( G ) of a graph G is the maximum number of G . � [ n ] K G V ; E ), where V = n;r = ( r E = { ( A; B ) : A ∩ B = ∅} : A graph-theoretic point of view The independence number n − 1 r � n= 2, then � ( K G n;r ) = r − 1 pairwise disjoint vertices of Kneser’s graph � , Erd˝ os–Ko–Rado � � If . A. Raigorodskii (MIPT, YND) 2017 Shanghai 3 / 11

� ( G ) of a graph G is the maximum number of G . � [ n ] K G V ; E ), where V = n;r = ( r E = { ( A; B ) : A ∩ B = ∅} : A graph-theoretic point of view The independence number n − 1 r � n= 2, then � ( K G n;r ) = r − 1 pairwise disjoint vertices of � ( G ) of a graph is the smallest number of colors needed Kneser’s graph � , Erd˝ os–Ko–Rado � � If . The chromatic number to color all the vertices so that no two vertices of the same color are joined by an edge. A. Raigorodskii (MIPT, YND) 2017 Shanghai 3 / 11

� ( G ) of a graph G is the maximum number of G . � [ n ] K G V ; E ), where V = n;r = ( r E = { ( A; B ) : A ∩ B = ∅} : A graph-theoretic point of view The independence number n − 1 r � n= 2, then � ( K G n;r ) = r − 1 pairwise disjoint vertices of � ( G ) of a graph is the smallest number of colors needed Kneser’s graph � , r � n= 2, then � ( K G n − 2 r + 2. n;r ) = Erd˝ os–Ko–Rado � � If . The chromatic number to color all the vertices so that no two vertices of the same color are joined by an edge. Lov´ asz, 1978 If A. Raigorodskii (MIPT, YND) 2017 Shanghai 3 / 11

Random subgraphs of Kneser’s graphs A. Raigorodskii (MIPT, YND) 2017 Shanghai 4 / 11

p ∈ [0 ; 1]. Then K G K G n;r ;p is obtained from n;r by keeping any of the edges p . Random subgraphs of Kneser’s graphs Let of Kneser’s graph with probability A. Raigorodskii (MIPT, YND) 2017 Shanghai 4 / 11

p ∈ [0 ; 1]. Then K G K G n;r ;p is obtained from n;r by keeping any of the edges p . n − 1 r � 2 is fixed and n → ∞ , then w.h.p. � ( K G n;r ; 1 = 2 ) ∼ r − 1 Random subgraphs of Kneser’s graphs Let of Kneser’s graph with probability Bogoliubskiy, Gusev, Pyaderkin, A.M., 2013 � � If . A. Raigorodskii (MIPT, YND) 2017 Shanghai 4 / 11

p ∈ [0 ; 1]. Then K G K G n;r ;p is obtained from n;r by keeping any of the edges p . n − 1 r � 2 is fixed and n → ∞ , then w.h.p. � ( K G n;r ; 1 = 2 ) ∼ r − 1 Random subgraphs of Kneser’s graphs Let of Kneser’s graph with probability Bogoliubskiy, Gusev, Pyaderkin, A.M., 2013 � � If . That’s another kind of stability. Moreover A. Raigorodskii (MIPT, YND) 2017 Shanghai 4 / 11

p ∈ [0 ; 1]. Then K G K G n;r ;p is obtained from n;r by keeping any of the edges p . n − 1 r � 2 is fixed and n → ∞ , then w.h.p. � ( K G n;r ; 1 = 2 ) ∼ r − 1 Random subgraphs of Kneser’s graphs " > 0 and let r = r ( n ) be a natural number such that Let n − 1 n 1 = 3 ). Let r ( n ) = o ( p n; r ) = (( r + 1) log n − r log r ) = ( of Kneser’s graph with probability r − 1 n → ∞ , Bogoliubskiy, Gusev, Pyaderkin, A.M., 2013 n − 1 p � (1 + " ) p n; r ) ( � � If � ( K G . n;r ;p ) = r − 1 p � (1 − " ) p n; r ) : ( That’s another kind of stability. Moreover Bollob´ as, Narayanan, A.M., 2016 Fix a real number � � 2 � . Then as � � � �� 1 if P → 0 if A. Raigorodskii (MIPT, YND) 2017 Shanghai 4 / 11

p ∈ [0 ; 1]. Then K G K G n;r ;p is obtained from n;r by keeping any of the edges p . n − 1 r � 2 is fixed and n → ∞ , then w.h.p. � ( K G n;r ; 1 = 2 ) ∼ r − 1 Random subgraphs of Kneser’s graphs " > 0 and let r = r ( n ) be a natural number such that Let n − 1 n 1 = 3 ). Let r ( n ) = o ( p n; r ) = (( r + 1) log n − r log r ) = ( of Kneser’s graph with probability r − 1 n → ∞ , Bogoliubskiy, Gusev, Pyaderkin, A.M., 2013 n − 1 p � (1 + " ) p n; r ) ( � � If � ( K G . n;r ;p ) = r − 1 p � (1 − " ) p n; r ) : ( That’s another kind of stability. Moreover Bollob´ as, Narayanan, A.M., 2016 Fix a real number � � 2 � . Then as � � � �� 1 if P → 0 if Successively improved by Das and Tran. A. Raigorodskii (MIPT, YND) 2017 Shanghai 4 / 11

Random subgraphs of Kneser’s graphs A. Raigorodskii (MIPT, YND) 2017 Shanghai 5 / 11

K G n;r is not so stable as the independence � ( K G < n − 2 r + 2 = � ( K G : n;r ) n;r ; 1 = 2 ) Random subgraphs of Kneser’s graphs Very simply the chromatic number of number: w.h.p. even However A. Raigorodskii (MIPT, YND) 2017 Shanghai 5 / 11

K G n;r is not so stable as the independence � ( K G < n − 2 r + 2 = � ( K G : n;r ) n;r ; 1 = 2 ) Random subgraphs of Kneser’s graphs n; r ; p , w.h.p. Very simply the chromatic number of � ( K G � ( K G n − 2 r + 2 : n;r ;p ) ∼ n;r ) = number: w.h.p. even However Kupavskii, 2016 For many different A. Raigorodskii (MIPT, YND) 2017 Shanghai 5 / 11

K G n;r is not so stable as the independence � ( K G < n − 2 r + 2 = � ( K G : n;r ) n;r ; 1 = 2 ) Random subgraphs of Kneser’s graphs n; r ; p , w.h.p. Very simply the chromatic number of � ( K G � ( K G n − 2 r + 2 : n;r ;p ) ∼ n;r ) = number: w.h.p. even g ( n ) is any growing function and r is arbitrary in the range n g ( n ), then for any fixed p , However � ( K G � ( K G : n;r ;p ) ∼ n;r ) Kupavskii, 2016 For many different For example, if between 2 and 2 − A. Raigorodskii (MIPT, YND) 2017 Shanghai 5 / 11