# Breaking Symmetries in Graphs

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1 Graph Packing p.1/18 Breaking Symmetries in Graphs Hemanshu Kaul kaul. Illinois Institute of Technology

2 Graph Packing p.2/18 Graphs and Colorings Graphs model binary relationships. In a graph G = (V (G),E(G)), the objects under study are represented by vertices included in V(G). If two objects are related then their corresponding vertices, say u and v in V (G), are joined by an edge that is represented as uv in E(G).

3 Graph Packing p.2/18 Graphs and Colorings For a university semester, we could define a conflict graph on courses, where each course is a vertex, and edges occur between pairs of vertices corresponding to courses with overlapping time. Then we could be interested in assigning rooms (colors) to the courses (vertices), such that a particular room is not assigned to two courses with overlapping times (vertices joined by an edge get different colors). The least number of rooms (colors) that would get the job done is called the chromatic number of the graph, denoted χ(g).

4 Graph Packing p.2/18 Graphs and Colorings Let G = (V (G),E(G)) be a graph. A proper k-coloring of G is a labeling of V (G) with k labels such that adjacent vertices get distinct labels. Chromatic Number, χ(g), is the least k such that G has a proper k-coloring.

5 Graph Packing p.2/18 Graphs and Colorings Some examples of Graphs: K n : Complete graph on n vertices. Each of the ( n) 2 pairs of vertices is joined by an edge. χ(k n ) = n P n : Path on n vertices. χ(p n ) = 2 C n : Cycle on n vertices. χ(c 2k ) = 2,χ(C 2k+1 ) = 3 K n1,n 2,...,n t : Complete t-partite graph on n 1 + n n t vertices. χ(k n1,n 2,...,n t ) = t

6 Graph Packing p.3/18 Symmetries in a Graph In K n, it is impossible to distinguish between any two vertices, u and v, because they are structurally identical. More formally, there is an bijection on V (K n ) that interchanges u and v without affecting the structure of K n. Such bijections are called automorphisms of G.

7 Graph Packing p.3/18 Symmetries in a Graph An automorphism of G is ρ : V (G) V (G), a bijection that preserves edges and non-edges of G, i.e., uv E(G) iff ρ(u)ρ(v) E(G). Aut(G) is the set (group) of all automorphisms of G. Aut(K n ) = S n, the Symmetric group formed by all the permutations on n objects. Aut(C n ) = D 2n, the Dihedral group formed by rotations and flips.

8 Graph Packing p.4/18 Distinguishing Vertices If we want to distinguish vertices in K n, we have to give each of them a distinct name (label). So, K n needs n labels. But, many times we can get away with using far less number of labels.

9 Graph Packing p.4/18 Distinguishing Vertices Suppose you have a key ring with n identical looking keys. You wish to label the handles of the keys in order to tell them apart. How many labels will you need? We want to figure out how many labels we need to distinguish vertices in C n.

10 Distinguishing Vertices Graph Packing p.4/18

11 Graph Packing p.4/18 Distinguishing Vertices C 3, C 4, and C 5 need three labels. When n 6, C n needs only two labels!! So we want to be able to decode the real identity of a vertex using only these (few) labels and the structure of the graph.

12 Graph Packing p.5/18 Distinguishing Number A distinguishing k-labeling of G is a labeling of V (G) with k labels such that the only color-preserving automorphism of G is the identity. Distinguishing Number, D(G), is the least k such that G has a distinguishing k-labeling. Introduced by Albertson and Collins in Since then, a whole class of research literature combining graphs and group actions has arisen around this topic.

13 Graph Packing p.5/18 Distinguishing Number Some examples: D(G) = 1 if and only if Aut(G) = {identity} D(K n ) = D(K 1,n ) = n. Both have Aut(G) = S n. It is possible to construct a graph G with Aut(G) = S n and D(G) = n. D(K n,n ) = n + 1. D(C n ) equals 3 if 3 n 5, and equals 2 if n 6. D(P n ) = 2.

14 Graph Packing p.5/18 Distinguishing Number In general the value of the Distinguishing number is strongly influenced by the relevant Automorphism group, rather than the particular graph. For a group Γ, D(Γ) = {D(G) : Aut(G) = Γ, G graph} Theorem [Albertson + Collins, 1996] D(D 2n ) = {2} unless n = 3, 4, 5, 6, 10, in which case, D(D 2n ) = {2, 3}. Theorem [Tymoczko, 2004] D(S n ) {2, 3,...,n}. Conjecture [Klavzar+ Wong + Zhu, 2005] D(S n ) = { n 1/k : k Z + }.

15 Graph Packing p.6/18 Distinguishing through proper colorings Distinguishing numbers tend to be fixed numbers that depend more on the automorphism structure than the graph structure. We want a proper coloring (not just an unrestricted labeling) that breaks all the symmetries of a graph, identifying each of its vertices uniquely.

16 Distinguishing through proper colorings Distinguishing numbers tend to be fixed numbers that depend more on the automorphism structure than the graph structure. We want a proper coloring (not just an unrestricted labeling) that breaks all the symmetries of a graph, identifying each of its vertices uniquely. Recall the conflict graph for courses. Find a coloring of the conflict graph that uniquely identifies each course as well as specifying the room each would use. We not only decode the real identity of a vertex using only these (few) labels and the structure of the graph, but get a useful partition of the vertices into conflict-free subsets. Graph Packing p.6/18

17 Graph Packing p.7/18 Distinguishing Chromatic Number A distinguishing proper k-coloring of G is a proper k-coloring of G such that the only color-preserving automorphism of G is the identity. Distinguishing Chromatic Number, χ D (G), is the least k such that G has a distinguishing proper k-coloring. Introduced by Collins and Trenk in 2005.

18 Graph Packing p.7/18 Distinguishing Chromatic Number A distinguishing proper k-coloring of G is a proper k-coloring of G such that the only color-preserving automorphism of G is the identity. Distinguishing Chromatic Number, χ D (G), is the least k such that G has a distinguishing proper k-coloring. Introduced by Collins and Trenk in Note that the chromatic number, χ(g), is an immediate lower bound for χ D (G).

19 Graph Packing p.8/18 Examples Not Distinguishing

20 Graph Packing p.8/18 Examples Not Distinguishing

21 Graph Packing p.8/18 Examples χ D (P 2n+1 ) = 3 and χ D (P 2n ) = 2 Distinguishing

22 Graph Packing p.8/18 Examples χ D (P 2n+1 ) = 3 and χ D (P 2n ) = 2 Distinguishing Not Distinguishing

23 Graph Packing p.8/18 Examples χ D (P 2n+1 ) = 3 and χ D (P 2n ) = 2 Distinguishing Distinguishing χ D (C n ) = 3 except χ D (C 4 ) = χ D (C 6 ) = 4

24 Graph Packing p.9/18 Motivating Question When are D(G) and χ D (G) small? Just one more than the minimum allowed?

25 Graph Packing p.9/18 Motivating Question When are D(G) and χ D (G) small? Just one more than the minimum allowed? Find a large general class of graphs for which D(G) χ D (G) χ(g) + 1 Our answer will be in terms of Cartesian Product of Graphs.

26 Graph Packing p.10/18 Cartesian Product of Graphs Let G = (V (G),E(G)) and H = (V (H),E(H)) be two graphs. G¾H denotes the Cartesian product of G and H. V (G¾H) = {(u,v) u V (G),v V (H)}. vertex (u,v) is adjacent to vertex (w,z) if either u = w and vz E(H) or v = z and uw E(G).

27 Graph Packing p.10/18 Cartesian Product of Graphs Let G = (V (G),E(G)) and H = (V (H),E(H)) be two graphs. G¾H denotes the Cartesian product of G and H. V (G¾H) = {(u,v) u V (G),v V (H)}. vertex (u,v) is adjacent to vertex (w,z) if either u = w and vz E(H) or v = z and uw E(G). Extend this definition to G 1 ¾G 2 ¾...¾G d. Denote G d = ¾ d i=1 G. A very special but important case, K d 2 denoted by Q d, is the d-dimensional hypercube.

28 Graph Packing p.10/18 Cartesian Product of Graphs G H G H

29 Graph Packing p.10/18 Cartesian Product of Graphs A graph G is said to be a prime graph if whenever G = G 1 ¾G 2, then either G 1 or G 2 is a singleton vertex. Prime Decomposition Theorem [Sabidussi(1960) and Vizing(1963)] Let G be a connected graph, then G = G p 1 1 ¾G p 2 2 ¾...¾G p d d, where G i and G j are distinct prime graphs for i j, and p i are constants. Theorem [Imrich(1969) and Miller(1970)] All automorphisms of a cartesian product of graphs are induced by the automorphisms of the factors and by transpositions of isomorphic factors.

30 Cartesian Product of Graphs Fact: Let G = ¾ d i=1 G i. Then χ(g) = max i=1,...,d {χ(g i)} Let f i be an optimal proper coloring of G i, i = 1,...,d. Canonical Coloring f d : V (G) {0, 1,...,t 1} as f d (v 1,v 2,...,v d ) = d i=1 f i (v i ) mod t, t = max{χ(g i )} i There is an edge between v and v in G if and only if they differ in only one coordinate, say v i and v i. So, v iv i is an edge in G i. Then f d (v) f d (v ) = f i (v i ) f i (v i) which is nonzero modulo t because f i is a proper coloring. Thus, f d gives a proper coloring ¾ of d i=1g i. Graph Packing p.10/18

31 Graph Packing p.11/18 Small Distinguishing Number Theorem [Bogstad + Cowen, 2004] D(Q d ) = 2, for d 4, and D(Q 2 ) = D(Q 3 ) = 3 where Q d is the d-dimensional hypercube. Theorem [Albertson, 2005] D(G d ) = 2, for d 4, if G is a prime graph. Theorem [Klavzar + Zhu, 2006] D(G d ) = 2, for d 3. Follows from D(K d n) = 2, proved using a probabilistic argument (when automorphisms of G have few fixed points then D(G) is large).

32 Graph Packing p.12/18 Large Distinguishing Chromatic Number Recall, χ(g) χ D (G) In general, χ D (G) might need many more colors than χ(g). Theorem [Collins + Trenk, 2006] χ D (G) = n(g) G is a complete multipartite graph. χ D (K n1,n 2,...,n t ) = t i=1 n i while χ(k n1,n 2,...,n t ) = t, arbitrarily far apart. Making our task more difficult.

33 Graph Packing p.13/18 Hamming Graphs and Hypercubes Theorem [Choi + Hartke + Kaul, 2006+] Given t i 2, χ D (¾ d i=1 K t i ) max i {t i } + 1, for d 5.

34 Graph Packing p.13/18 Hamming Graphs and Hypercubes Theorem [Choi + Hartke + Kaul, 2006+] Given t i 2, χ D (¾ d i=1 K t i ) max i {t i } + 1, for d 5. Corollary : Given t 2, χ D (K d t ) t + 1, for d 5. Both these upper bounds are 1 more than their respective lower bounds.

35 Graph Packing p.13/18 Hamming Graphs and Hypercubes Theorem [Choi + Hartke + Kaul, 2006+] Given t i 2, χ D (¾ d i=1 K t i ) max i {t i } + 1, for d 5. Corollary : Given t 2, χ D (K d t ) t + 1, for d 5. Both these upper bounds are 1 more than their respective lower bounds. These results allow us to exactly determine the distinguishing chromatic number of hypercubes. Corollary : χ D (Q d ) = 3, for d 5.

36 Graph Packing p.14/18 Main Theorem Theorem [Choi + Hartke + Kaul, 2006+] Let G be a graph. Then there exists an integer d G such that for all d d G, χ D (G d ) χ(g) + 1. By the Prime Decomposition Theorem for Graphs, G = G p 1 1 ¾G p 2 2 ¾...¾G p k k, where G i are distinct prime graphs. (This prime decomposition can be found in polynomial time) Then, d G = max i=1,...,k {lg n(g i) p i } + 5 Note, n(g) = (n(g 1 )) p 1 (n(g 2 )) p 2 (n(g k )) p k At worst, d G = lg n(g) + 5 suffices.

37 Graph Packing p.14/18 Main Theorem Theorem [Choi + Hartke + Kaul, 2006+] Let G be a graph. Then there exists an integer d G such that for all d d G, χ D (G d ) χ(g) + 1. d G = max i=1,...,k {lg n(g i) p i } + 5 when, n(g) = (n(g 1 )) p 1 (n(g 2 )) p 2 (n(g k )) p k d G is unlikely to be a constant, as the example of Complete Multipartite Graphs indicates pushing χ D (K n1,n 2,...,n t ) down from n(g) to t + 1 can not happen with only a fixed number of products.

38 Graph Packing p.15/18 Proof Idea for Main Theorem Fix an optimal proper coloring of G. Embed G in a complete multipartite graph H. Form H by adding all the missing edges between the color classes of G. Now work with H. BUT G H χ D (G) χ D (H)!

39 Graph Packing p.15/18 Proof Idea for Main Theorem Fix an optimal proper coloring of G. Embed G in a complete multipartite graph H. Form H by adding all the missing edges between the color classes of G. Then construct a distinguishing proper coloring of H d that is also a distinguishing proper coloring of G d. Study Distinguishing Chromatic Number of Cartesian Products of Complete Multipartite Graphs.

40 Graph Packing p.16/18 Complete Multipartite Graphs Theorem [Choi + Hartke + Kaul, 2006+] Let H be a complete multipartite graph. Then χ D (H d ) χ(h) + 1, for d lg n(h) + 5. This is already enough to prove Theorem 1 for prime graphs.

41 Graph Packing p.16/18 Complete Multipartite Graphs Theorem [Choi + Hartke + Kaul, 2006+] Let H be a complete multipartite graph. Then χ D (H d ) χ(h) + 1, for d lg n(h) + 5. This is already enough to prove Theorem 1 for prime graphs. Theorem [Choi + Hartke + Kaul, 2006+] Let H ¾ = k i=1 Hp i i, where H i are distinct complete multipartite graphs. Then χ D (H d ) χ(h) + 1, for d max i=1,...,k {lg n i p i } + 5, where n i = n(h i ).

42 Graph Packing p.17/18 Outline of the Proof for Hamming Graphs Start with the canonical proper coloring f d of cartesian products of graphs, f d : V (Kt d ) {0, 1,...,t 1} with f d (v) = d f(v i ) mod t, i=1 where f(v i ) = i is an optimal proper coloring of K t.

43 Graph Packing p.17/18 Outline of the Proof for Hamming Graphs Derive f from f d by changing the color of the following vertices from f d (v) to : Origin : Group 1 : A = d 2 i=1 A i, where A i = {e 1 i,j 1 + i j d + 1 i} v : the vertex with all coordinates equal to 1 except for the d + 1 th coordinate which equals 0. 2 e 1 i,j is the vertex with all coordinates equal to 0 except for the ith and jth coordinates which equal 1.

44 Graph Packing p.18/18 Outline of the Proof for Hamming Graphs Uniquely identify each vertex of K d t by reconstructing its original vector representation by using only the colors of the vertices and the structure of the graph.

45 Graph Packing p.18/18 Outline of the Proof for Hamming Graphs Uniquely identify each vertex of K d t by reconstructing its original vector representation by using only the colors of the vertices and the structure of the graph. Step 1. Distinguish v from the Origin and the Group 1 by counting their distance two neighbors in the color class. In Q 6, v has no vertices with color within distance two of it.

46 Graph Packing p.18/18 Outline of the Proof for Hamming Graphs Uniquely identify each vertex of K d t by reconstructing its original vector representation by using only the colors of the vertices and the structure of the graph. Step 2. Distinguish the Origin by counting the distance two neighbors in color class. In Q 6, Origin is the only vertex with color that is within distance two of every other vertex of color.

47 Graph Packing p.18/18 Outline of the Proof for Hamming Graphs Uniquely identify each vertex of K d t by reconstructing its original vector representation by using only the colors of the vertices and the structure of the graph. Step 3. Assign the vector representations of weight one, with 1 as the non-zero coordinate, to the correct vertices. In Q 6, vertex has 5 neighbors in Group 1, has 4, has 3 (and is at distance 6 from v ), has 3 (and is at distance 4 from v ), has 2, and has 1 such neighbors.

48 Graph Packing p.18/18 Outline of the Proof for Hamming Graphs Uniquely identify each vertex of K d t by reconstructing its original vector representation by using only the colors of the vertices and the structure of the graph. Step 4. Assign the vector representations of weight one, with k > 1 as the non-zero coordinate, to the correct vertices, by recovering the original canonical colors of all the vertices.

49 Graph Packing p.18/18 Outline of the Proof for Hamming Graphs Uniquely identify each vertex of K d t by reconstructing its original vector representation by using only the colors of the vertices and the structure of the graph. Step 5. Assign the vector representations of weight greater than one to the correct vertices. Let x be a vertex with weight ω 2. Then x is the unique neighbor of the vertices, y 1,y 2,...,y ω, formed by changing exactly one non-zero coordinate of x to zero that is not the Origin. For example, in Q 6, is the unique vertex with and as its only weight one neighbors, and so on.

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