Edges and Triangles Po-Shen Loh Carnegie Mellon University Joint - PowerPoint PPT Presentation

Edges and Triangles Po-Shen Loh Carnegie Mellon University Joint work with Jacob Fox

Edges in triangles Observation There are graphs with the property that every edge is contained in a triangle, but no edge is in more than one triangle.

Edges in triangles Observation There are graphs with the property that every edge is contained in a triangle, but no edge is in more than one triangle.

Edges in triangles Observation There are graphs with the property that every edge is contained in a triangle, but no edge is in more than one triangle. Question (Erd˝ os-Rothschild) What if the total number of edges must be at least 0 . 001 n 2 ? Must some edge be in many triangles?

Regularity Lemma Szemer´ edi Regularity Lemma For every ǫ , there is M such that every graph can be ǫ -approximated by an object of complexity bounded by M . Triangle Removal Lemma For any ǫ , there is a δ such that every graph with ≤ δ n 3 triangles can be made triangle-free by deleting only ǫ n 2 edges.

Regularity Lemma Szemer´ edi Regularity Lemma For every ǫ , there is M such that every graph can be ǫ -approximated by an object of complexity bounded by M . Triangle Removal Lemma For any ǫ , there is a δ such that every graph with ≤ δ n 3 triangles can be made triangle-free by deleting only ǫ n 2 edges. Dependency between parameters: (From Regularity Lemma.) 1 δ is tower of height power of 1 ǫ . (Fox.) 1 δ is tower of height logarithmic in 1 ǫ .

Lower bound for Erd˝ os-Rothschild Observation Let c be a constant. Given cn 2 edges, each of which is in a triangle, there is always an edge which is in log ∗ n triangles.

Lower bound for Erd˝ os-Rothschild Observation Let c be a constant. Given cn 2 edges, each of which is in a triangle, there is always an edge which is in log ∗ n triangles. Proof: If the graph has over δ n 3 triangles, then double-counting already gives an edge in at least 3 δ n 3 cn 2 triangles.

Lower bound for Erd˝ os-Rothschild Observation Let c be a constant. Given cn 2 edges, each of which is in a triangle, there is always an edge which is in log ∗ n triangles. Proof: If the graph has over δ n 3 triangles, then double-counting already gives an edge in at least 3 δ n 3 cn 2 triangles. Else, Removal Lemma gives ǫ n 2 edges hitting all the triangles.

Lower bound for Erd˝ os-Rothschild Observation Let c be a constant. Given cn 2 edges, each of which is in a triangle, there is always an edge which is in log ∗ n triangles. Proof: If the graph has over δ n 3 triangles, then double-counting already gives an edge in at least 3 δ n 3 cn 2 triangles. Else, Removal Lemma gives ǫ n 2 edges hitting all the triangles. Every edge is in a triangle, so total number of triangles ≥ cn 2 3 . Then some edge is in at least cn 2 / 3 triangles. ǫ n 2

Lower bound for Erd˝ os-Rothschild Observation Let c be a constant. Given cn 2 edges, each of which is in a triangle, there is always an edge which is in log ∗ n triangles. Proof: If the graph has over δ n 3 triangles, then double-counting already gives an edge in at least 3 δ n 3 cn 2 triangles. Else, Removal Lemma gives ǫ n 2 edges hitting all the triangles. Every edge is in a triangle, so total number of triangles ≥ cn 2 3 . Then some edge is in at least cn 2 / 3 triangles. ǫ n 2 Either case gives an edge in at least min { 3 δ n c , c 3 ǫ } triangles.

Lower bound for Erd˝ os-Rothschild Observation Let c be a constant. Given cn 2 edges, each of which is in a triangle, there is always an edge which is in log ∗ n triangles. Proof: If the graph has over δ n 3 triangles, then double-counting already gives an edge in at least 3 δ n 3 cn 2 triangles. Else, Removal Lemma gives ǫ n 2 edges hitting all the triangles. Every edge is in a triangle, so total number of triangles ≥ cn 2 3 . Then some edge is in at least cn 2 / 3 triangles. ǫ n 2 Either case gives an edge in at least min { 3 δ n c , c 3 ǫ } triangles. δ = √ n , and 1 ǫ = power of log ∗ n . Take 1 �

Previous work Theorem (Alon-Trotter) For any constant c < 1 4 , there is a cn 2 -edge graph with every edge in a triangle, but the most popular edge only in √ n triangles.

Previous work Theorem (Alon-Trotter) For any constant c < 1 4 , there is a cn 2 -edge graph with every edge in a triangle, but the most popular edge only in √ n triangles. Theorem (Edwards; Khadˇ ziivanov-Nikiforov) 4 n 2 edges, there is always one in ≥ 1 Given any 1 6 n triangles.

Previous work Theorem (Alon-Trotter) For any constant c < 1 4 , there is a cn 2 -edge graph with every edge in a triangle, but the most popular edge only in √ n triangles. Theorem (Edwards; Khadˇ ziivanov-Nikiforov) 4 n 2 edges, there is always one in ≥ 1 Given any 1 6 n triangles. Theorem (Bollob´ as-Nikiforov) 4 n 2 − o ( n 1 . 4 ) edges, each of which is in a triangle, there Given any 1 is always some edge in at least n 4 / 5 triangles.

Previous work Theorem (Alon-Trotter) For any constant c < 1 4 , there is a cn 2 -edge graph with every edge in a triangle, but the most popular edge only in √ n triangles. Theorem (Edwards; Khadˇ ziivanov-Nikiforov) 4 n 2 edges, there is always one in ≥ 1 Given any 1 6 n triangles. Theorem (Bollob´ as-Nikiforov) 4 n 2 − o ( n 1 . 4 ) edges, each of which is in a triangle, there Given any 1 is always some edge in at least n 4 / 5 triangles. Question (Erd˝ os, 1987) Given cn 2 edges, each of which is in a triangle, is there always some edge which is in at least n ǫ triangles, for a constant ǫ ?

New result Theorem (Fox, L.) There are n -vertex graphs with n 2 � 1 − e − (log n ) 1 / 6 � 4 edges, each of which is in a triangle, but with no edge in more than n 14 / log log n triangles.

New result Theorem (Fox, L.) There are n -vertex graphs with n 2 � 1 − e − (log n ) 1 / 6 � 4 edges, each of which is in a triangle, but with no edge in more than n 14 / log log n triangles. Remarks: Every edge is in under n o (1) triangles. The edge density approaches 1 4 from below. Sharp transition: after edge density 1 4 , some edge is in a linear number of triangles.

Construction materials Theorem (Hoeffding-Azuma) For any L -Lipschitz random variable X determined by n independent samples, t 2 P [ | X − E [ X ] | > t ] ≤ 2 e − 2 L 2 n .

Construction materials Theorem (Hoeffding-Azuma) For any L -Lipschitz random variable X determined by n independent samples, t 2 P [ | X − E [ X ] | > t ] ≤ 2 e − 2 L 2 n . Corollary: If a coin is flipped n times, the probability that the 2 ± √ n is at least a constant. number of heads falls within n

Construction materials Theorem (Hoeffding-Azuma) For any L -Lipschitz random variable X determined by n independent samples, t 2 P [ | X − E [ X ] | > t ] ≤ 2 e − 2 L 2 n . Corollary: If a coin is flipped n times, the probability that the 2 ± √ n is at least a constant. number of heads falls within n Classical result In even dimensions d , the Euclidean ball of radius r has = π d / 2 r d � B ( d ) � Vol ( d / 2)! . r

Construction 0 Core tripartite graph: Take 3 copies of the lattice cube of side r in dimension d = r 5 . C 0 r A B 0 r 0 r

Construction 0 Core tripartite graph: Take 3 copies of the lattice cube of side r in dimension d = r 5 . C 0 r A B 0 r 0 r Let µ be the expected squared-distance between two random points in a single cube. A – B edges correspond to squared-distances in µ ± d .

Construction 0 Core tripartite graph: Take 3 copies of the lattice cube of side r in dimension d = r 5 . C 0 r A B 0 r 0 r Let µ be the expected squared-distance between two random points in a single cube. A – B edges correspond to squared-distances in µ ± d . A – C edges correspond to squared-distances in µ 4 ± 2 d . B – C edges correspond to squared-distances in µ 4 ± 2 d .

Properties C 0 r A B 0 r 0 r Positive edge density: The edge density between A and B is the probability that two random points in the cube have squared-distance within µ ± d .

Properties C 0 r A B 0 r 0 r Positive edge density: The edge density between A and B is the probability that two random points in the cube have squared-distance within µ ± d . The squared-distance between u = ( u 1 , . . . , u d ) and v = ( v 1 , . . . , v d ) is the sum of independent ( u i − v i ) 2 , each ranging between 0 and r 2 .

Properties C 0 r A B 0 r 0 r Positive edge density: The edge density between A and B is the probability that two random points in the cube have squared-distance within µ ± d . The squared-distance between u = ( u 1 , . . . , u d ) and v = ( v 1 , . . . , v d ) is the sum of independent ( u i − v i ) 2 , each ranging between 0 and r 2 . The typical deviation from µ is r 2 √ d = r 4 . 5 ≪ d , since d = r 5 , so the A – B edge density approaches 1!

Properties C 0 r A B 0 r 0 r Every A–B edge is in a triangle: A – B endpoints have squared-distance µ ± d . Their integer-rounded midpoint has squared-distance µ 4 ± 2 d from each endpoint.

Properties Every A–B edge is in few triangles: Given 0 = (0 , . . . , 0) and z = ( z 1 , . . . , z d ) with � z � 2 = µ ± d . 2 , . . . , z d 2 + a d Consider points x = ( z 1 2 + a 1 2 )