densities of cliques and independent sets in graphs
DESCRIPTION
Densities of cliques and independent sets in graphs. Yuval Peled , HUJI. Joint work with Nati Linial , Benny Sudakov , Hao Huang and Humberto Naves. High level motivation. How can we study large graphs? Approach: Sample small sets of vertices and examine the induced subgraphs . - PowerPoint PPT PresentationTRANSCRIPT
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Densities of cliques and independent sets in graphs
Yuval Peled, HUJI
Joint work with Nati Linial, Benny Sudakov, Hao Huang and Humberto Naves.
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High level motivation How can we study large graphs?
Approach: Sample small sets of vertices and examine the induced subgraphs.
What graph properties can be inferred from its local profile?
What are the possible local profiles of large graphs?
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Local profiles of graphs What are the possible local profiles of
(large) graphs?
For graphs H,G, we denote by d(H;G) the induced density of H in G, i.e.
d(H;G):= The probability that |H| random vertices in G induce a copy of H.
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Definition: Given a family of graphs, is the set of all such that , a sequence of graphs with
and
Problem: Characterize this set.
Local profiles of graphs
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Characterizing seems to be a hard task: A mathematical perspective:
Many hard problems fall into this framework. E.g. for t=1, the problem is equivalent to computing the
inducibility of graph,
a parameter known only for a handful of graphs. A computational perspective:
[Hatami, Norine 11’]: Satisfiability of linear inequalities in is undecidable.
Local profiles of graphs
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The case of two cliques is already of interest: Turan’s Theorem:
Kruskal-Katona Theorem: (r<s)
Minimize subject to this constraint? much harder: solved only recently for r=2:
Razborov 08’ (s=3), Nikiforov 11’ (s=4 ,(Reiher (arbitrary s)
Local profiles of two cliques
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Motivation - quantitative versions of Ramsey’s theorem: Investigate distributions of monochromatic cliques in a
red/blue coloring of the complete graph. Goodman’s inequality:
The minimum is attained by G(n,½), conjectured by Erdos to minimize
for every r. Refuted by Thomasson for every r>3.
A clique and an anticlique
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A consequence from Goodman’s inequality:
[Franek-Rodl 93’] The analog of this is false for r=4, by a blow up of the following graph: V = {0,1}^13, v~u iff dist(v,u) ∈ {1,4,5,8,9,11}
Fundamental open problem: Find graphs with few cliques and anticliques.
We are interested in the other side of
A clique and an anticlique (II)
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How big can both d(Ks;G) and
d(Kr;G) be?
Many cliques and anticliques
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What graphs has many cliques and anticliques?Example: r=s=3.
Many cliques and anticliques
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First guess:A clique on some fraction of the vertices
Second guess:Complements of these graphs
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t 1-t
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Main theoremLet r, s > 2. Suppose that and let q be
the unique root in [0,1] of Then,
Namely, given the maximum of is attained in one of two graphs: a clique on a fraction of the vertices, or the complement of such graph.
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More theorems Stability: such that every
sufficiently large graph G with
is close to the extremal graph. Max-min:
where
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Proof of main theorem Strategy:I. Reduce the problem to threshold graphs.II. Reformulate the problem for threshold
graphs as an optimization problem.III. Characterize the solutions of the
optimization problem.
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Proof of main theorem Strategy:I. Reduce the problem to threshold graphs.II. Reformulate the problem for threshold
graphs as an optimization problem.III. Characterize the solutions of the
optimization problem.
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Given a graph G and vertices u,v the shift of G from u to v is defined by the rule: Every other vertex w with w~u and w≁v gets
disconnected from u and connected to v. A graph G with V=[n] is said to be shifted if
for every i<j the shift of G from j to i does not change G.
Fact: Every graph can be made shifted by a finite number of shifting operations.
Shifting in a nutshell
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Lemma: Shifting does not decrease the number of s-cliques in the graph.
Proof: Consider the shift from j to i. If a subset C of V forms a clique in G and not in the shifted graph S(G), then C \ {j} U {i} forms a clique in S(G) and not in G.
Cor: By symmetry, shifting does not decrease the number of r-anticliques.
Shifting cliques
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Def: A graph is called a threshold graph if there is an order on the vertices, such that every vertex is adjacent to either all or none of its predecessors.
Lemma: A shifted graph is a threshold graph. Proof: Consider the following order:
Cor: The extremal graph is a threshold graph.
Threshold graphs
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Proof of main theorem Strategy:I. Reduce the problem to threshold graphs.II. Reformulate the problem for threshold
graphs as an optimization problem.III. Characterize the solutions of the
optimization problem.
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Every threshold graph G can be encoded as a point in
Threshold graphs
A_1 A_2 A_3 A_4 A_2k-1 A_2k
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The densities are (upto o(1)):densities in threshold graphs
A_1 A_2 A_3 A_4 A_2k-1 A_2k
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The new form of our optimization problem is:
We need to prove that every maximum is either supported on x_1,y_1 or on y_1,x_2.
Optimization problem
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It suffices to show that for every a,b>0, the maximum of
is either supported on x_1,y_1 or on y_1,x_2.
Why? For both problems have the same set of
maximum points.
Optimization problem
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Proof of main theorem Strategy:I. Reduce the problem to threshold graphs.II. Reformulate the problem for threshold
graphs as an optimization problem.III. Characterize the solutions of the
optimization problem.
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Let k,r,s≥2 be integers, a,b>0 reals, and
the polynomials defined above. Then,every non-degenerate maximum of
is either supported on x_1,y_1 or on y_1,x_2.
(x,y) is non-degenerate if the zeros in the sequence (y_1,x_2,y_3,…,x_k,y_k) form a suffix.
Technical lemma
A_1 A_2 A_3 A_4 A_2k-1 A_2kA_1 U A_3
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Proof Let (x,y) be a non-degenerate maximum of f:
, otherwise we can increase f by a perturbation that increases the smaller element.
WLOG x_1>0, otherwise x exchange roles with y, and p with q (by looking at the complement graph).
We show that x_3=y_2=x_2=0.
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Proof (x_3=0) Define the following matrices:
If x_3>0 and (x,y) is non-degenerate then B is positive definite.
For , let x’ be defined by
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Proof (x_3=0) (II)
Then,
If A is singular – choose Av=0, v≠0. If A is invertible – choose
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Proof (x_3=0) (III) Hence,
contradicting the maximality of f(x,y).
Proving y_2=0, x_2=0 is done with similar methods.
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For the max-min theorem: Consider
(a=b=1).
For r=s=3, Goodman inequality and our bound completely determine the set
Stability – obtained using Keevash’s stable Kruskal-Katona theorem.
Remarks
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?
The End
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For l≤m,
Hence,
and