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

Distinguishing Infinite Graphs Anthony Bonato 2 Dedicated to the memory of Michael Albertson

Distinguishing Infinite Graphs Anthony Bonato 3

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

5

Distinguishing number let G be 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 6

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

6-cycle Distinguishing Infinite Graphs Anthony Bonato 8 D(C 6 ) = 2

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

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 10

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(Q n ) = 2 if n ≥ 4, otherwise = 3 (Klavžar, Wong, Zhu, 05): D(G) ≤ Δ(G) unless G is a clique, regular biclique, or C 5, 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 11

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

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

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 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: The infinite random graph For all finite A B there exists z

Distinguishing Infinite Graphs Anthony Bonato 16 R as a limit graph fix R 0 a finite graph suppose R t is defined to form R t+1, for each finite set S in R t, add a vertex z s joined to each vertex of S and to no other vertices of R t the limit graph is e.c. so isomorphic to R RtRt S zszs

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?

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

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 19

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 Distinguishing Infinite Graphs Anthony Bonato 20 age(R) is closed under unions, but age(P) is not

The infinite case (Imrich, Klav ž ar, Trofimov, 07): –D(R) = 2 –D(Q k ) = 2 if k is an infinite cardinal –D(G) ≤ Δ(G) (Watkins, Zhou, 07): if G is a locally finite tree with 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 21

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 22

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 23

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: Distinguishing Infinite Graphs Anthony Bonato 24 T z x y a structure S is weak e.c. iff its graph G(S) is weak e.c.

Examples of weak e.c. structures R, homogeneous K n -free graphs, Henson digraphs, homogenous k-uniform hypergraphs, ARO, … (B, Delić, 04) There are 2 א 0 many non- isomorphic weak e.c. undirected graphs. Distinguishing Infinite Graphs Anthony Bonato 25

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 26

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 Distinguishing Infinite Graphs Anthony Bonato 27 xy Z R B

Proof continued if f is an automorphism of G, then –f fixes B (one-way rays are asymmetric) –if f(x) = y for x,y red, 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 28

Proof continued process inductively all pairs {x i, y i } of vertices from V(G) all pairs initially unprocessed induction-step n+1: consider {x n+1, y n+1 } delete z n+1 from unprocessed pairs and relabel Distinguishing Infinite Graphs Anthony Bonato 29 x n+1 y n+1 ZnZn znzn z n+1 by weak e.c.

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 30

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 i th neighbourhoods preserving colours determine D(G(n,p)) for all p = p(n) Distinguishing Infinite Graphs Anthony Bonato 31

Distinguishing Infinite Graphs Anthony Bonato 32

Distinguishing Infinite Graphs Anthony Bonato 33 preprints, reprints, contact: Google: “Anthony Bonato”