1. 1 Universality, Tolerance, Chaos and Order Noga Alon, Tel Aviv University Szemerédi’s Conference Budapest, August 2010

2. 2 Universality and Tolerance

3. 3 Universal Graphs Definition: H - A family of graphs. G is H - universal if it contains every member of H as a subgraph Example: G= is H -universal for the family of all 2-regular graphs on 7 vertices

4. 4 Objective: Construct sparse H-universal graphs for interesting families H Motivation: VLSI circuit design

5. 5 Some early results: Sparse universal graphs for forests Chung, Graham and Pippenger (78) Friedman and Pippenger (87) Sparse universal graphs for planar graphs and graphs with small separators Bondy (71) Rödl (81) Babai,Chung,Erdős and Graham (82) Capalbo (99) Capalbo and Kosaraju (99)

6. 6 In these cases the number of edges is linear or nearly linear This is not the case for general cubic graphs

7. 7 Universal graphs for bounded-degree graphs: H (k,n)=all graphs on n vertices, max-degree ≤ k Question: Estimate the minimum possible number of edges of an H (k,n)-universal graph, and of one on n vertices A,Capalbo,Kohayakawa,Rödl,Ruciński,Szemerédi: Universality and Tolerance (00): Ω(n 2-2/k ) edges are needed, O(n 2-1/k log 1/k n) suffice (using more than n vertices) O(n 2-c/k log k ) edges suffice (with n vertices)

8. 8 Moreover: O(n 2-1/2k log 1/2k n) edges suffice for a fault tolerant universal graph: every set of 1% of its edges contains all bipartite graphs on n+n vertices with maximum degree at most k ACKRRS (01): O(n 2-2/k log 1+8/k n) edges suffice (using more than n vertices) A and Asodi (02): For k=3, n vertices, O(n 1.87.. ) edges suffice Dellamonica,Kohayakawa,Rödl and Ruciński (08): n vertices, O(n 2-1/2k log 1/k n) edges suffice

9. 9 A+ Capalbo (08): Theorem 1: For all k ≥3 there is c=c(k) and an explicit H (k,n)-universal G on n vertices with at most c n 2-2/k log 4/k n edges. Theorem 2: For all k ≥ 3 there is c=c(k) and an explicit H (k,n)-universal G with at most c n 2-2/k edges.

10. 10 The proof of Theorem 1 is probabilistic, based on the rapid mixing of random walks on expanders. The proof of Theorem 2 applies properties of high-girth expanders, and provides a deterministic embedding procedure. The proof for even k is simpler, the one for odd k requires an additional effort: a new graph decomposition result.

11. 11 Theorem 2 for k=4: The minimum possible number of edges of a graph that contains a copy of every graph on n vertices with maximum degree at most 4 is Θ(n 3/2 )

12. 12 The lower bound: Simple counting: there are “many” 4-regular graphs on n vertices, and a graph with m edges cannot contain too many subgraphs with 2n edges The upper bound: Construction using high-girth expanders

13. 13 The construction: Let a,d be absolute constants, put m=a n 1/2, and let F be a d-regular Ramanujan expander of girth at least ⅔ log d-1 m. Thus all nontrivial eigenvalues of F are of absolute value at most 2(d-1) 1/2. Define G=(V,E), where V=(V(F)) 2 and (a 1,a 2 ) is adjacent to (b 1,b 2 ) iff a i and b i are within distance 2 in F for i=1 and/or i=2. Clearly |E|=O(n 3/2 ). Main claim: G is H (4,n)-universal.

14. 14 A homomorphism from a graph Z to a graph T is a mapping of V(Z) to V(T) such that adjacent vertices in Z are mapped to adjacent vertices in T. Thus there is an injective homomorphism from Z to T iff Z is a subgraph of T. P n - the path of length n. A homomorphism from P n to F is a walk on F. The k-th power T k of a graph T is the graph on V(T) in which two vertices are adjacent iff their distance in T is at most k.

15. 15 Let H be a graph on n vertices with maximum degree at most 4. By Petersen’s Theorem H can be decomposed into two spanning subgraphs H 1,H 2, each having max. degree at most 2. There are bijective homomorphisms g i from H i to P n 2, To embed H in G we define homomorphisms f i from P n to F so that f(v)=(f 1 (g 1 (v)), f 2 (g 2 (v)) ) is an injective homomorphism from H to G. This is done by defining each f i as an appropriate non- back-tracking walk on F.

16. 16 The existence of the required walks f i is proved using the spectral properties of the expander F and the fact it has high girth. The construction of universal graphs for H (k,n) with k>4 even is similar The odd case requires more efforts

17. 17 A graph is thin if every connected component of it is either a subgraph of a cycle with pendant edges or a graph with max. degree 3 and at most two vertices of degree 3

18. 18 Fact: Every thin graph can be mapped homomorphically and bijectively to the forth power of a path. Theorem: Let H be a graph of maximum degree k. Then there are k thin spanning subgraphs H 1, H 2, …,H k of H, so that each edge of H lies in two of the graphs H i.

19. 19 A universal graph for H (k,n): Let F be a high-girth Ramanujan graph on m=a n 1/k vertices. Construct G=(V(G),E(G)) as follows: V(G)=(V(F)) k (a 1,a 2, …,a k ) and (b 1,b 2, …,b k ) are adjacent iff there are at least two indices i so that a i and b i are within distance 4 in F.

20. 20 Open: Is there an H (k,n)-universal graph on n vertices with O(n 2—2/k ) edges ?

21. 21 Motzkin: The essence of Ramsey Theory is that complete chaos is impossible: every sufficiently large system contains a substantial ordered one. Chaos and Order

22. 22

23. 23 Chvátal, Rödl, Szemerédi and Trotter (83): The Ramsey number of any bounded degree graph is linear in its number of vertices, that is: for any fixed k, there is a constant c=c(k) so that for any graph G with maximum degree k and n vertices, any red-blue coloring of the edges of the complete graph on c n vertices contains a monochromatic copy of G. The proof is based on Szemerédi’s Regularity Lemma

24. 24 Kohayakawa, Rödl, Schacht and Szemerédi (2010): The complete graph can be replaced by a sparser (random) graph, with only O(n 2-1/k log 1/k n) edges. Moreover: any red-blue coloring of such a graph contains a monchromatic H (k,n)-universal graph. The proof combines the regularity method with a subtle embedding lemma Open: is there such a graph with only O(n 2-2/k ) edges ?

25. 25 Subgraph containment problems A and Marx (2010+): some of the techniques in the investigation of universal graphs for bounded degree graphs are relevant to the study of the complexity of subgraph containment problems The colored H-subgraph problem: given a fixed graph H on the vertices {1,2, …, h}, decide if an input graph G with n vertices colored by 1,2,..,h contains a copy of H respecting the coloring.

26. 26 Example H= G=

27. 27 Freuder (90), A,Yuster and Zwick (95): It can be also solved in time 2 O(h) n O(w), where w is the tree-width of H. Clearly: The H-colored subgraph problem can be solved in time O(n h ) Marx (07): Assuming the exponential time hypothesis, it cannot be solved in time n o(w/ log w) The exponential time hypothesis [ Impagliazzo, Paturi and Zane (01) ]: Satisfiability on m variables cannot be solved in time 2 o(m).

28. 28 A rough sketch of the proof of Marx: Represent the satisfiability formula by a graph F with O(m) edges Call a function mapping each vertex of F to a connected subset of H an embedding of depth d (of F into H) if the endpoints of each edge of F are mapped to sets that are within distance 1 in H, and the inverse image of each vertex of H is of size at most d Show, crucially, that if the tree-width of H is w, then there is an embedding of depth at most O( m log w /w ) (of F into H).

29. 29 Construct a graph G by replacing each vertex i of H by a set of vertices, colored i, corresponding to all 2 O( m log w/ w) possible assignments of the variables mapped to i, and define the edges of G to ensure that satisfying assignments will correspond to copies of H. If we can now solve the colored H–subgraph problem in time n o( w / log w ), we’ll solve satisfiability in time [2 O(m log w / w) ] o(w / log w) = 2 o(m), contradicting the exponential time hypothesis. ■

30. 30 It thus follows that to improve the hardness result to a tight n o(w) bound, it will suffice to show that any graph F with O(m) edges admits an embedding of depth at most O( m /w) into any graph of tree width w. A and Marx (2010+): This is false, namely, the log w extra term cannot be omitted. The proof is by showing that certain balanced homomorphisms of random cubic graphs into expanders do not exist. The study of those applies similar ideas to those used in the investigation of universal graphs.

31. 31 This is meant to stay somewhat obscure Happy Birthday, Endre !