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List Coloring and Euclidean Ramsey

Theory

Noga Alon, Tel Aviv U.

Bertinoro, May 2011

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Euclidean Ramsey Theory

The Hadwiger-Nelson Problem: what is theminimum number of colors required to colorthe points of the Euclidean plane with no twopoints of distance 1 with the same color ?

Equivalently: what is the chromatic number ofthe unit distance graph in the plane ?

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Nelson (1950): at least 4

Isbell (1950): at most 7

Both bounds have also been proved byHadwiger (1945)

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By the Erdős de-Bruijn Theorem it suffices toconsider finite subgraphs of the unit distance graph

The lower bound (4) and the upper bound (7) have not been improved since 1945

Wormald, O’donnell: The unit distance graph contains subgraphs of arbitrarily high girth andchromatic number 4

The higher dimensional analogs have been considered as well (Frankl and Wilson)

Surveys: Chilakamarri, Soifer

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Erdős, Graham, Montgomery, Rothschild, Spencer and Straus (73,75,75):For a finite set K in the plane, let HK be thehypergraph whose set of vertices is R2, wherea set of |K| points forms an edge iff it is an isometric copy of K.

Problem: What’s the chromatic number x(HK) of HK ?

(The case |K|=2 is the Hadwiger-Nelson problem)

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Fact (EGMRSS): If K is the set of 3 vertices ofan equilateral triangle, then x(HK)=2

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Conjecture 1 (EGMRSS): For any set of 3 points KX(HK) ≤3

Fact (EGMRSS): If K is the set of 3 vertices ofan equilateral triangle, then x(HK)=2

Conjecture 2 (EGMRSS): For any set of 3 points Kwhich is not the set of vertices of an equilateraltriangle, x(HK) ≥ 3.

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List Coloring [ Vizing (76), Erdős, Rubin and Taylor(79) ]

Def: G=(V,E) - graph or hypergraph, the listchromatic number xL(G) is the smallest kso that for every assignment of lists Lv foreach vertex v of G, where |Lv|=k for all v,there is a coloring f of V satisfying f(v) ε Lv

for all v, with no monochromatic edge

Clearly x(G) ≥ xL(G) for all G, strict inequality is possible

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Fact 1: Deciding if for an input graph G, x(G) ≤ 3 is NP-complete

Fact 2 (ERT): Deciding if for an input graph G,xL(G) ≤ 3 is Π2-complete

Question 1: xL(Unit Distance Graph)=?

Question 2: For a given finite K in the plane, xL(HK)=?

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New [ A+Kostochka (11) ]: For any finite K in the plane xL(HK) is infinite !

That is: for any finite K in the plane and for anypositive integer s, there is an assignment of a listof s colors to any point of the plane, such that in any coloring of the plane that assigns to each point a color from its list, there is a monochromaticisometric copy of K

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The reason is combinatorial:

Thm 1 (A-00): For any positive integer s there isa finite d=d(s) such that for any (simple, finite) graph G with average degree at least d, xL(G)>s.

Thm 2 (A+Kostochka): For any positive integersr,s there is a finite d=d(r,s) such that for anysimple (finite) r-uniform hypergraph H with average (vertex)- degree at least d, xL(H)>s.

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A hypergraph is simple if it contains no two edgessharing more than one common vertex.

E.g., for r=4, the following is not allowed

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Note: this provides a linear time algorithm todistinguish between a simple (hyper-)graphwith list chromatic number at most s, and onewith list chromatic number at least b(s).

E.g., distinguishing between a graph G with xL(G)≤3 and one with xL(G)≥1000 is easy.

There is no such known result for usual chromaticchromatic number, and it is unlikely that such aresult holds (Dinur, Mosell, Regev)

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Deriving the geometric result from the combinatorial one:Given a finite K in the plane, prove, by inductionon d, that there exists a finite, simple d-regular hypergraph Hd whose vertices are points in the plane in which every edge is an isometric copy of K.

Hd is obtained by taking |K| copies of Hd-1,obtained according to a random rotation of K.

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Example: |K|=3, Hd

is obtained from 3copies of Hd-1

Hd-1

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The proofs of the combinatorial results are probabilistic.

Theorem 1 (graphs with large average degreehave high list chromatic number) is proved byassigning to each vertex, randomly andindependently, a uniform random s-subset ofthe set [2s]={1,2,…,2s}.

It can be shown that with high probability thereis no proper coloring using the lists, providedthe average degree is sufficiently large.

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Note: if the graph is a complete bipartite graphwith d vertices in each vertex class, where d isarbitrarily large, there are at least s2d potentialproper colorings (with colors in {1,2,..,s} to thefirst vertex class, and colors in {s+1,s+2,…,2s} tothe second). Each such potential coloring willcome from the lists with probability 1/22d, hencethe expected number of colorings from the listsis at least (s/2)2d which is far bigger than 1 !

This means that a naïve computation does not suffice.

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The idea is to first expose the lists for a randomly chosen set of a 1/d1/2 - fraction of the vertices, andthen show that if d is sufficiently large then with high probability, no coloring from these lists can be extended to a proper coloring of the whole graph using the lists of the other vertices.

More precisely:

G contains a subgraph H with minimum degree atleast d/2.

In this subgraph, let W be a random set containinga 1/d1/2 fraction of the vertices.

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Expose the random lists of the vertices in W. Ifd>20s, say, then with high probability, at least half of the vertices of H do not belong to W and for each s-subset S of [2s], have a neighbor in W whose list is S.

Let n be the number of vertices of H, and let A be a set of half of them as above.

For each fixed coloring f of the vertices of W fromtheir lists, and for each vertex v in A, there are atleast s+1 colors used by f for the neighbors of v. Thus, if the list of v is contained in these s+1 colors, there will not be any proper extension of f to v.

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It follows that for each fixed coloring f of W fromthe lists, the probability that W can be extended to a proper coloring of all vertices in A using theirlists is at most [(1-(s+1)/4s ]n/2.

Therefore, the probability that there exists an f that can be extended to a proper coloring of H from the lists is at most

3|W| [(1-(s+1)/4s]n/2 < exp (n/d1/2 log 3-(n/2)(s+1) /4s) <1.

■

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The proof of Theorem 2 (simple r-uniform hypergraphs with high average degree have highlist-chromatic number) proceeds by induction on r.

The induction hypothesis has to be strengthened:it is shown that if the average degree is high, thenthere is an assignment of s-lists so that in anycoloring using the lists, a constant fraction of alledges is monochromatic.

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This requires a decomposition result: 99% of theedges of any r-uniform hypergraph with large average degree can be decomposed into edge-disjoint subhypergraphs, each having large minimum degree which is at least 1/r times its average degree.

The probabilistic estimates are strong enough toapply simultaneously to all subhypergraphs, usingthe same random choice.

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Open Problems

The Hadwiger-Nelson Problem: x(Unit distance graph in the plane)= ?

EGMRSS: Is it true that for any non-equilateraltriangle K, x(HK)=3 ?

Ronsenfeld: Is x(Odd distance graph in the plane) finite ?

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What is the smallest possible estimate for d=d(r,s)in Theorem 2 ? (simple r-uniform hypergraphs with average degree at least d have list chromatic number bigger than s). Is it rΘ(s) ?

Is there an efficient (deterministic) algorithm that finds, for a given input simple (hyper-)graphwith sufficiently large minimum degree, lists Lv,each of size s, for the vertices, so that there is noproper coloring using the lists ?

Can one give such lists and a witness showingthere is no proper coloring using them ?

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Is probability essential in proofs of this type ?