A reverse Sidorenko inequality Independent sets, colorings, and graph homomorphisms

Transcription

1 A reverse Sidorenko inequality Independent sets, colorings, and graph homomorphisms Yufei Zhao (MIT) Joint work with Ashwin Sah (MIT) Mehtaab Sawhney (MIT) David Stoner (Harvard)

2 Question Fix d. Which d-regular graph G maximizes! " #/% &? # independent sets Question 2 Fix d and q. Which d-regular graph G maximizes ' ( " #/% &? Question 3 # proper q-colorings Fix d and H. Which d-regular graph G maximizes hom(",.) #/% &? # graph homomorphisms

3 Independent sets:! " = hom ", Colorings: ( ) " = hom(", + ) ) Widom Rowlinson model: hom(", )

4 Independent sets Question. Fix d. Which d-regular graph G maximizes! " #/% &? Asked by Granville in 988 at Banff in an effort to resolve the Cameron Erdős conjecture on the number of sum-free subsets of {,, n} Conjectured maximizer: K d,d Alon (99) proved an asymptotic version (' ) Kahn (200) proved the conjecture for bipartite G via entropy method Z. (200) removed the bipartite hypothesis via bipartite swapping trick! " *!(". * ) Theorem (Kahn + Z.). Let G be an n-vertex d-regular graph. Then! "!. 0,0 2/(*0) = 2 05# 2/(*0) Davies, Jenssen, Perkins & Roberts (207) gave a new proof using a novel occupancy method, which found applications in sphere packing and spherical codes [Jenssen, Joos, Perkins 208]

5 Graph homomorphisms Question 3. Fix d and H. Which d-regular graph G maximizes hom(%, ') )/+,? [Galvin, Tetali 2004] Among bipartite graphs, G = K d,d is the maximizer (extending [Kahn 0]) Q. Can the bipartite hypothesis be dropped? [Z. 20] Yes for certain families of H, such as threshold graphs (generalizing independent sets). H = K q (q-colorings) remained open The bipartite hypothesis cannot always be dropped. E.g., H =, maximizer is K d+, not K d,d. [Cohen, Perkins, Tetali 207] Widom Rowlinson model (H = [Sernau 207] ': maximizer is neither K d,d nor K d+ ): G = K d+ is the maximizer Open: Among 3-regular graphs, is there a finite list of possible maximizers G for hom(%, ') )/+,? (We only know that this list is more than just K 3,3 and K 4 )

6 Graph homomorphisms Question 3. Fix d and H. Which d-regular graph G maximizes hom(%, ') )/+,? Wide open in general (see my survey Extremal regular graphs) Conjecture (Davies, Jenssen, Perkins, Roberts 207). For all fixed H, among triangle-free G, G = K d,d is always the maximizer (already known for bipartite G [Galvin, Tetali 2004])

7 Independent sets in irregular graphs d u = degree of u in G Degree-degree distribution: probab. distribution of (d u, d v ) for uniformly random edge uv Question. Given the degree-degree distribution, which G maximizes! " #/% &? e.g., 20% edges have endpoint degrees (3,4), 30% edges Conjecture (Kahn 0). Maximizer is a disjoint union of complete bipartite graphs We prove this conjecture Theorem (Sah, Sawhney, Stoner, Z., 8+). Let G be a graph without isolated vertices. Then! " (!. /0,/ 2 #/(/ 0 / 2 ) )% +(&) Independent sets are biclique-maximizing Conjecture (Galvin 06). An analogous inequality for hom ", 6 (False; which G and H?)

8 Proper colorings Question 2. Fix d and q. Which d-regular graph G maximizes! " # $/& '? Conjectured answer: K d,d [Galvin, Tetali 04] True for bipartite G [Davies, Jenssen, Perkins, Roberts 8] True for d = 3 & [Davies] d = 4 (computer-assisted) We prove the conjecture Theorem (Sah, Sawhney, Stoner, Z. 8++). Let q N and G an n-vertex d-regular graph. Then! " #! " +,,,./(0,) Theorem (Sah, Sawhney, Stoner, Z.). Let q N and G a graph without isolated vertices. Then! " # 2 3& 4(')! " +,5,, 6 $/(, 5, 6 ) Proper colorings are biclique-maximizing

9 The number of independent sets and proper q-colorings satisfies % & ( )*,(.) % 0 2, 4 5/( 2 4 ) f counts independent sets or proper q-colorings

10 Graph homomorphisms Question 3. Fix d and H. Which d-regular graph G maximizes hom(%, ') )/+,? Conjecture (Davies, Jenssen, Perkins, Roberts 7). Among triangle-free G, G = K d,d is always the maximizer (already known for bipartite G [Galvin, Tetali 04]) We prove this conjecture Theorem (Sah, Sawhney, Stoner, Z.). Let G be a triangle-free n-vertex d-regular graph. Then hom(%, ') hom(. /,/, ') 0/(/) Theorem (SSSZ). Let G be a triangle-free graph without isolated vertices. Then hom(%, ') (,) hom(. /6,/ 7, ') )/(/ 6/ 7 ) Always biclique-maximizing among triangle-free graphs False for every G with a triangle! Counterexample: ' = : : ε as : 0

11 Reverse Sidorenko inequality Sidorenko s conjecture: for bipartite G, all H! ", $! & ', $ ( ) [Hatami] [Conlon, Fox, Sudakov] [Li, Szegedy] [Kim, Lee, Lee] [Conlon, Kim, Lee, Lee] [Szegedy] [Conlon, Lee] Open for G = K 5,5 \ C 0 (Möbius strip) Our result: for triangle-free d-regular G! ", $! & +,+, $ (())/+/! ", $ = hom ", $ /H $? ) 0 )! ", 3/(()) (Hatami s graph norm ; [Conlon, Lee]). For graphon 4: 0, ' [0,], 4 ;/ 4 ) 4 ;<,<? bipartite G (Sidorenko s conjecture) triangle-free d-regular G (our result) Theorem (Sah, Sawhney, Stoner, Z.). Let G be a triangle-free graph and 4: 0, ' [0,]. Then!(", 4) = >? A()) 4 ;< B,<C

12 Reverse Sidorenko inequality Given!: Ω $ Ω & R, e.g.,! )*,, = x x 2 y y 2 y 3 $/5 -! 0 $, $! 0 $, &! 0 $, 2! 0 &, $! 0 &, &! 0 &, 2 30 $ 30 & 3 $ 3 & 3 2. * /., * Theorem (Sah, Sawhney, Stoner, Z.). Triangle-free graph G = (V, E) without isolated vertices :! 78 (0 7, 0 8 ) 3= > 6 78 :! 78 )@ A,@B 5 f 5 Ω f 2 : Ω Ω 2 R 0 Ω 5 Ω 2 f 45 Ω Ω f f

13 Reverse Sidorenko inequality Theorem (Sah, Sawhney, Stoner, Z.). Triangle-free graph G = (V, E) without isolated vertices! " 2 (6, 6 2 ) 89 : " 2 +; <,;= Graphical analogs of Brascamp Lieb type inequalities: 5 f 5 Ω f 2 : Ω Ω 2 R 0 Ω 5 Ω 2 f 45 Ω Ω f f ! " # " % " # ' () " % ' ( * Note that (by Hölder) " +,,. " ',. Future direction: extensions to simplicial complexes

14 The number of independent sets and proper q-colorings % & ( )*,(.) % 0 2, 4 5/( 2 4 ) f counts independent sets or proper q-colorings

15 The number of proper list colorings Strong induction hypothesis (example):

16 Proof strategy: Induction = + By induction +

17 Proof strategy: Reduction to local inequality + Remains to show

18 Proof strategy: Reduction to local inequality + Remains to show

19 Proof strategy: Reduction to local inequality By Cauchy Schwarz (more generally Hölder):!" + $%! + $ " + % Remains to show

20 Proof strategy: Reduction to local inequality + + Remains to show Break inequality into two parts: top & bottom

21 Proof strategy: Local inequality = + Remains to show Break inequality into two parts: top & bottom

22 Proof strategy: Local inequality This is a minimal instance of the inequality Remains to show Much more difficult if G has triangles (not always true for other models!)

23 A useful matrix inequality Define the mixed l p,q norm of matrix A = (a ij ) by first taking l p norm of each row, and then taking l q norm of the results, i.e.! ",$ & & ) '( " $ " + $ ' ( Lemma. For positive semidefinite (PSD) matrix A with nonneg entries, and q, Question. Is it true that for all p q,,! +,$! +,+! $,$!, ",$! ","! $,$?

24 Graph homomorphisms Question 3. Fix d and H. Which d-regular graph G maximizes hom(%, ') )/+,? Let H be a nonneg weighted graph (model) hom(%, ') = partition function of some stat. phys. model, e.g., hard-core, Ising, Potts H is biclique-maximizing if - % : = hom(%, ') satisfies )/( ) - % , (,) H is clique-maximizing if - % : = hom(%, ') satisfies Our results: H = - % + 9(,) :) )/(6 8 :)) i.e., conditioned on degree-degree distribution i.e., conditioned on degree distribution (indep sets) and K q (proper colorings) are both biclique-maximizing More generally, every partially looped K q (semiproper colorings) is biclique-maximizing

25 Ferromagnetism and anti-ferromangnetism Given a nonneg weighted graph/model H, we say that H is ferromagnetic if its edge-weight matrix is positive semidefinite, i.e., all eigenvalues are nonnegative: 0 $ % $ & $ ' (e.g., H = ) H is antiferromagnetic if its edge-weight matrix has at most one positive eigenvalue: $ % $ & 0 $ ' (e.g., indep sets and colorings) Theorem (Sah, Sawhney, Stoner, Z.). Every ferromagnetic model is clique-maximizing Conjecture. Every clique-maximizing model is ferromagnetic Conjecture 2. Every antiferromagnetic model is biclique-maximizing Our results verify Conj. 2 for independent sets and colorings. Open for Potts model Widom Rowlingson model is clique-maximizing among d-regular G [Cohen, Perkins, Tetali] but not for irregular G, and it is not ferromagnetic.

26 Two-spin systems! " An Ising model with nonneg edge-weight matrix " # is ferromagnetic if!# " % and antiferromagnetic if!# " % E.g., independent set is antiferromagnetic 0 Corollary (Sah, Sawhney, Stoner, Z.). A 2-spin model is Biclique-maximizing if antiferromagnetic, and Clique-maximizing if ferromagnetic This generalizes the result for independent sets A similar classification for 3-spin systems is currently out of reach

27 Summary of main results Independent sets and proper colorings are biclique-maximizing Every ferromagnetic model is clique-maximizing Every model is biclique-maximizing when restricted to triangle-free graphs Reverse Sidorenkoinequality (Sah, Sawhney, Stoner, Z.). Triangle-free graph G = (V, E) without isolated vertices! " #$ & ' #$ () #, ) $ ),-. " #$ & ' #$ 0 2,3 Corollary. For triangle-free G without isolated vertices, H hom(8, 9) " #$ &(:) hom(; <3,< 2, 9) =/(< 3< 2 ) Conjecture. Every antiferromagnetic model is biclique-maximizing (e.g., Potts). 5 f 45 4 f 5 Ω Ω 5 Ω 4 f 34 Ω 3 f 2 : Ω Ω 2 R 0 Ω 2 3 f 23 2