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List C o l o r i n g and Euclidean Ramsey Theory

List C o l o r i n g and Euclidean Ramsey Theory. Noga Alon, Tel Aviv U. Bertinoro, May 2011. TexPoint fonts used in EMF. Read the TexPoint manual before you delete this box.: A A A A. Euclidean Ramsey Theory.

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List C o l o r i n g and Euclidean Ramsey Theory

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  1. List Coloring and Euclidean Ramsey Theory Noga Alon, Tel Aviv U. Bertinoro, May 2011 TexPoint fonts used in EMF. Read the TexPoint manual before you delete this box.: AAAA

  2. 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 ?

  3. Nelson (1950): at least 4 Isbell (1950): at most 7 Both bounds have also been proved byHadwiger (1945)

  4. 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

  5. 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, where a 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)

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

  7. Fact (EGMRSS): If K is the set of 3 vertices ofan equilateral triangle, then x(HK)=2 Conjecture 1 (EGMRSS): For any set of 3 points K X(HK) ≤3 Conjecture 2 (EGMRSS): For any set of 3 points Kwhich is not the set of vertices of an equilateraltriangle, x(HK) ≥ 3.

  8. 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 Lvfor all v, with no monochromatic edge Clearly x(G) ≥ xL(G) for all G, strict inequality is possible

  9. 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)=?

  10. 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

  11. 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.

  12. 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

  13. 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 chromatic chromatic number, and it is unlikely that such a result holds (Dinur, Mosell, Regev)

  14. 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.

  15. Hd-1 Example: |K|=3, Hdis obtained from 3copies of Hd-1

  16. 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.

  17. Note: if the graph is a complete bipartite graphwith d vertices in each vertex class, where d isarbitrarily large, there are at least s2dpotentialproper 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.

  18. The idea is to first expose the lists for a randomlychosen 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.

  19. 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.

  20. It follows that for each fixed coloring f of W fromthe lists, the probability that W can be extendedto 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. ■

  21. 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.

  22. 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.

  23. 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 ?

  24. 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) algorithmthat 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 ?

  25. Is probability essential in proofs of this type ?

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