Distinguishing Infinite Graphs

1. Discrete Mathematics Days 2009 May 23, 2009 Distinguishing Infinite Graphs Anthony Bonato Ryerson University Distinguishing Infinite Graphs Anthony Bonato

2. Dedicated to the memory of Michael Albertson 1946 - 2009 Distinguishing Infinite Graphs Anthony Bonato

3. Distinguishing Infinite Graphs Anthony Bonato

4. Solution • f(139) = 2 • rephrase the question… • find the minimum number of colours on the vertices of the n-cycle Cn so that no automorphism preserves the colours Distinguishing Infinite Graphs Anthony Bonato

5. Distinguishing Infinite Graphs Anthony Bonato

6. Distinguishing number • letGbe a graph with n vertices • a d-distinguishing labelling of G is a vertex colouring (not necessarily proper) with d colours so that no automorphism of G preserves the colours • always a n-distinguishing labelling • the minimum d such that G is has a d-distinguishing labelling is the distinguishing number of G, written D(G) Distinguishing Infinite Graphs Anthony Bonato

7. 5-cycle • D(C5) ≤ 3 • D(C5) > 2 by direct checking Distinguishing Infinite Graphs Anthony Bonato

8. 6-cycle • D(C6) = 2 Distinguishing Infinite Graphs Anthony Bonato

9. Infinite 2-way path, P D(P) = 2 Distinguishing Infinite Graphs Anthony Bonato

10. Asymmetric graphs • asymmetric graphs:no non-trivial automorphisms • G asymmetric D(G) = 1 • the distinguishing number is a measure of asymmetry Distinguishing Infinite Graphs Anthony Bonato

11. Distinguishing finite graphs • (Albertson, Collins, 96): • If Aut(G) is abelian, then D(G) ≤ 2 • If Aut(G) is dihedral, then D(G) ≤ 3 • (Russell, Sundaram, 98): complexity of computing D(G) in the class AM (= Arthur-Merlin) • (Bogstad, Cowen, 04): D(Qn) = 2 if n ≥ 4, otherwise = 3 • (Klavžar, Wong, Zhu, 05): D(G) ≤ Δ(G) unless G is a clique, regular biclique, or C5, where D(G) ≤ Δ(G) + 1 • (Cheng, 06): computing D(G) is O(nlogn) if G is acylic • (Albertson, Boutin, 07): Kneser graphs are 2-distinguishable Distinguishing Infinite Graphs Anthony Bonato

12. Application: robotics • (Lynch, 01) robotic manipulation, such as throwing, catching, and controlling the orientation of a rolling ball Distinguishing Infinite Graphs Anthony Bonato

13. G(n,p)(Erdős, Rényi, 63) • n a positive integer, p = p(n) in (0,1) • G(n,p): probability space on graphs with nodes {1,…,n}, two nodes joined independently and with probability p 4 1 2 3 5 Distinguishing Infinite Graphs Anthony Bonato

14. Random graphs are rigid • an event A holds in G(n,p)asymptotically almost surely (a.a.s.) if A holds with probability tending to 1 as n → ∞ Theorem (Erdős, Rényi, 63) If 1- ln n/n ≥ p ≥ ln n/n, then a.a.s. G(n,p) is asymmetric (i.e. D(G(n,p)) = 1) Distinguishing Infinite Graphs Anthony Bonato

15. The infinite random graph Theorem (Erdős,Rényi, 63): With probability 1, any two graphs sampled from G(N,p) are isomorphic. • isotype R unique with the e.c. property: B For all finite A there exists z Distinguishing Infinite Graphs Anthony Bonato

16. R as a limit graph • fix R0a finite graph • suppose Rtis defined • to form Rt+1, for each finite set S in Rt, add a vertex zs joined to each vertex of S and to no other vertices of Rt • the limit graph is e.c. so isomorphic to R Rt S zs Distinguishing Infinite Graphs Anthony Bonato

17. Homogeneity • R is homogeneous: isomorphism between finite induced subgraphs extend to automorphism • R is vertex-transitive, edge-transitive, … • R plays a prominent role in the (Lachlan, Woodrow, 80) classification of the countable homogeneous graphs • so D(R) > 1 … but what is it? Distinguishing Infinite Graphs Anthony Bonato

18. Structures • signatureS = (si: i in N): sequence of positive integers • structureG with signature S: nonempty set V(S), along with relations of arity si on V(G) • examples • graphs, digraphs, orders • hypergraphs • unary predicates Distinguishing Infinite Graphs Anthony Bonato

19. Structures, continued • induced substructures, isomorphisms, homogeneity, distinguishing number, etc generalize naturally from graphs to structures • given a structure S, its graph, written G(S), has vertices V(S), with x and y joined if x and y are in some relation of S Distinguishing Infinite Graphs Anthony Bonato

20. Ages and free amalgamation • age(G): set of isotypes of finite induced substructures • age(R) = all finite graphs • age(P) = unions of finite paths • age(G) has Free Amalgamation Property (FAP): class is closed under unions • age(R) is closed under unions, but age(P) is not Distinguishing Infinite Graphs Anthony Bonato

21. The infinite case • (Imrich, Klavžar, Trofimov, 07): • D(R) = 2 • D(Qk) = 2 if k is an infinite cardinal • D(G) ≤ Δ(G) • (Watkins, Zhou, 07): if G is alocally finite treewith no end-vertices, then D(G) = 2 • (Laflamme, Van Thé, Sauer, 09): • if G is a homogeneous structure whose age(G) has FAP, then D(G) = 2 Distinguishing Infinite Graphs Anthony Bonato

22. Conjecture • a structure G is primitive if there is no non-trivial equivalence relation on V(G) preserved by Aut(G) • eg, R is primitive, as is an infinite clique Conjecture(Laflamme, Van Thé, Sauer, 09) If G is a primitive structure, then D(G) is 2 or infinite. Distinguishing Infinite Graphs Anthony Bonato

23. Proving D(R)=2 • (Imrich, Klavžar, Trofimov, 07): ad hoc argument • (Laflamme, Van Thé, Sauer, 09): properties of permutation groups; fixing types • we develop a unified combinatorial approach via an adjacency property Distinguishing Infinite Graphs Anthony Bonato

24. Weak e.c. property • a graph G that is not a clique is weak-e.c. if for each pair x, y of (possibly equal) non-joined vertices and a finite set T of vertices, there is a vertex z joined to x and y but not joined nor equal to a vertex in T: y x T z • a structure S is weak e.c. iff its graph G(S) is weak e.c. Distinguishing Infinite Graphs Anthony Bonato

25. Examples of weak e.c. structures • R, homogeneous Kn-free graphs, Henson digraphs, homogenous k-uniform hypergraphs, ARO, … • (B, Delić, 04) There are 2א0many non-isomorphic weak e.c. undirected graphs. Distinguishing Infinite Graphs Anthony Bonato

26. D = 2 is ubiquitous Theorem (B, Delić, 09) If a countable relational structure S has the weak e.c. property, then D(S) ≤ 2. • gives alternative proof that D(R) = 2 (Imrich et al) and covers most cases of homogeneous graphs and digraphs (Laflamme et al) Distinguishing Infinite Graphs Anthony Bonato

27. Sketch of proof of theorem • let G = G(S) • G satisfies property (♣): there is an induced one-way path Z such that for all x,y not in Z, there is a z in Z joined to exactly one of x,y x y R B Z Distinguishing Infinite Graphs Anthony Bonato

28. Proof continued • if f is an automorphism of G, then • f fixes B (one-way rays are asymmetric) • if f(x) = y for x,yred, then contradiction by (♣) • remainder of the proof: show weak e.c. implies (♣) • Aut(S,R,B) is isomorphic to a subgroup of Aut(G(S),R,B) Distinguishing Infinite Graphs Anthony Bonato

29. Proof continued • process inductively all pairs {xi, yi} of vertices from V(G) • all pairs initially unprocessed • induction-step n+1: consider {xn+1, yn+1} • delete zn+1 from unprocessed pairs and relabel yn+1 xn+1 Zn by weak e.c. zn+1 zn Distinguishing Infinite Graphs Anthony Bonato

30. FAP Corollary (BD, 09) If G is a countable homogeneous structure whose age(G) has FAP, then D(G) = 2. • recovers (Laflamme et al, 09) result for homogenous structures with FAP Distinguishing Infinite Graphs Anthony Bonato

31. Directions • infinite structures G with D(G) = 2? • Urysohn space • distinguishing chromatic number of infinite graphs? • d-distinguishing labelling must be a proper colouring • i-local distinguishing number of infinite graphs? • minimum number of colours needed so no two vertices have isomorphic ith neighbourhoods preserving colours • determine D(G(n,p)) for all p = p(n) Distinguishing Infinite Graphs Anthony Bonato

32. Distinguishing Infinite Graphs Anthony Bonato

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