Distinguishing Chromatic Number and NG-Graphs

Transcription

1 Wellesley College Groups in Galway May 19, 2018

2 Table of contents 1 Distinguishing Number and Distinguishing Chromatic Number 2 Nordhaus-Gaddum Inequality 3 NG-Graphs

3 Distinguishing Number and Distinguishing Chromatic Number Graphs and Automorphisms A Graph G = (V, E) consists of a vertex set V and an edge set E consisting of unordered pairs of distinct vertices. An automorphism of graph G = (V, E) is a map f : V V that is 1-1, onto, and preserves edges. Examples: For graph G, the function f defined by f (a) = b, f (b) = a, f (c) = c and f (d) = d is an automorphism. For graph H, any function f : V V that is 1-1 and onto is an automorphism.

4 Distinguishing Number and Distinguishing Chromatic Number The Key Problem Given a ring of seemingly identical keys that open different doors, how many color tags are needed to distinguish the keys? Example: Not Distinguishing Distinguishing D(C 3 ) = 3

5 Distinguishing Number and Distinguishing Chromatic Number Distinguishing Number (Albertson and Collins 1996) Definition: The Distinguishing Number D(G) of a graph G is the minimum number of colors needed to assign a color to each vertex of G so that no nontrivial automorphism of G preserves colors. (Not necessarily a proper coloring.) Example: D(C 4 ) = 3.

6 Distinguishing Number and Distinguishing Chromatic Number The Distinguishing Number of Complements The complement of graph G = (V, E) is the graph G = (V, E). Thus xy E if and only if xy E Observe that D(G) = D(G) since aut(g) is the same as aut(g). K 3,2 K 3,2 D(K 3,2 ) = 3

7 Distinguishing Number and Distinguishing Chromatic Number A Scheduling Problem Given a set of events and information about which ones conflict, how many colors are needed to distinguish them and assign them to rooms? V = {events} E = {xy : events x and y conflict} A proper coloring of G = (V, E) gives an assignment of the events to rooms. If also distinguishing, we can uniquely identify each event.

8 Distinguishing Number and Distinguishing Chromatic Number Distinguishing Chromatic Number (Collins/Trenk 2006) Definition: The Distinguishing Chromatic Number χ D (G) of a graph G is the minimum number of colors needed to properly color V (G) so that no nontrivial automorphism of G preserves colors. Example: χ D (C 4 ) = 4. χ D (C 4 ) > 3 χ D (C 4 ) 4

9 Distinguishing Number and Distinguishing Chromatic Number An easy example: Paths

10 Distinguishing Number and Distinguishing Chromatic Number Distinguishing Number for Cycles Posed by Frank Rubin in J. Recreational Math, Construction for n 6: n D(C n ) D(C n ) = 2 for n 6

11 Distinguishing Number and Distinguishing Chromatic Number Distinguishing Chromatic Number for Cycles n χ D (C n ) χ D (C n ) 3 χ D (C n ) 3 for n 3 for n 7

12 Distinguishing Number and Distinguishing Chromatic Number Easy Bounds for χ D (G) max{d(g), χ(g)} χ D (G) D(G) χ(g) Proper Distinguishing Proper and Distinguishing χ(c 6 ) = 2 D(C 6 ) = 2 χ D (C 6 ) 4

13 Distinguishing Number and Distinguishing Chromatic Number Which graphs have χ D (G) = V (G)? χ D (K 5 ) = 5 χ D (K 5 ) = 5 χ D (C 4 ) = 4

14 Distinguishing Number and Distinguishing Chromatic Number Complete Multipartite Graphs A graph G is a complete multipartite graph if the vertex set can be partitioned as V (G) = V 1 V 2 V k so that the edge set of G is E(G) = {xy : x V i, y V j, i j}.

15 Distinguishing Number and Distinguishing Chromatic Number The graphs with χ D (G) = V (G) are... Theorem: A graph G has χ D (G) = V (G) if and only if G is a complete multipartite graph.

16 Distinguishing Number and Distinguishing Chromatic Number The Distinguishing Number for Complete Multipartite Graphs Must distinguish between vertices in the same partite set vertices in partite sets of the same size

17 Distinguishing Number and Distinguishing Chromatic Number The Distinguishing Number for Complete Multipartite Graphs is... Theorem: If G is a complete multipartite graph consisting of j i partite sets of size a i for distinct a i then ( ) p D(G) = max{p : j i for each i}. For K 2,2,2,2, a 1 = 2 j 1 = 4 Need ( p 2) 4, so p = 4. a i

18 Distinguishing Number and Distinguishing Chromatic Number Complements of Complete Multipartite Graphs Theorem: If G is a complete multipartite graph and G is its complement then D(G) = D(G) = χ D (G) G = K 2,2,2,2 G = K 3,2,1 D(G) = D(G) = χ D (G) = 4 D(G) = D(G) = χ D (G) = 3

19 Nordhaus-Gaddum Inequality Nordhaus-Gaddum Inequality from 1956 Theorem: If G is a graph on n vertices then 2 n χ(g) + χ(g) n + 1. Example: G = K n. Generalize for χ D? χ(k n ) = n χ(k n ) = 1

20 Nordhaus-Gaddum Inequality The Lower Bound for χ D Let G be a graph on n vertices, and recall that χ(g) χ D (G). Thus 2 n χ(g) + χ(g) χ D (G) + χ D (G) Examples exist to show this is tight. For example, any graph G with D(G) = 1 and 2 n χ(g) + χ(g).

21 Nordhaus-Gaddum Inequality The Upper Bound for χ D Theorem (Collins/Trenk 2013): If G is a graph on n vertices then χ D (G) + χ D (G) n + D(G). Example showing bound is tight: G = K n. χ D (K n ) = n χ D (K n ) = n D(G) = n More generally, if G is a complete multipartite graph, χ D (G) = n χ D (G) = D(G)

22 Nordhaus-Gaddum Inequality The Upper Bound for χ D Theorem (Collins/Trenk 2013): If G is a graph on n vertices then Proof: χ D (G) + χ D (G) n + D(G). Fix a distinguishing coloring of G using k = D(G) colors. Let V i be the set of vertices of color i. Consider the induced graphs H i = G[V i ] and H i = G[V i ].

23 Nordhaus-Gaddum Inequality The Upper Bound for χ D (continued) To show: χ D (G) + χ D (G) n + D(G). Proof (continued): Color H i and H i using V i + 1 shades of color i (by NG-inequality). This uses ( V i + 1) = V + k = n + D(G) colors. This coloring is proper and also distinguishing. Any automorphism that preserves colors maps V i V i, but the colorings of H i and H i are distinguishing.

24 NG-Graphs Nordhaus-Gaddum Graphs Definition: A graph G with n vertices is an NG-graph if it satisfies χ(g) + χ(g) = n + 1. Example: C 5 is an NG-graph since χ(c 5 ) + χ(c 5 ) = = Example: C 4 is not an NG-graph since χ(c 4 ) + χ(c 4 ) = <

25 NG-Graphs Previous Characterizations of NG-graphs Finck (1966): Characterized NG-graphs using arrays of vertices with certain required adjacencies and nonadjacencies. Starr and Turner (2008): G is an NG-graph iff its vertex set can be partitioned into three sets, {x}, S, T where x is a vertex, G[S] = K χ(g) 1 and G[T ] = K χ(g) 1 Neither leads to a polynomial time recognition algorithm.

26 NG-Graphs The ABC partition of a Graph Definition: For a graph G, the ABC-partition of G is A = A G = {v V (G) : deg(v) = χ(g) 1} B = B G = {v V (G) : deg(v) > χ(g) 1} C = C G = {v V (G) : deg(v) < χ(g) 1}. Example: The house graph with χ = 3.

27 NG-Graphs ABC Partitions for Several Graphs NG-Graphs non NG-graphs

28 NG-Graphs Our NG-graph Characterization Theorem (Collins/Trenk 2013): A graph G is an NG-graph if and only its ABC-partition satisfies (i) A and G[A] is a clique, an independent set, or C 5. (ii) G[B] is a clique. (iii) G[C] is an independent set. (iv) uv E(G) for all u A, v B. (v) uw E(G) for all u A, w C. A K a, all B I a or C 5 K b none? I c C

29 NG-Graphs Three types of NG-graphs An NG-graph G is an NG-1 graph if G[A] is a clique, (χ(g) = A + B ). an NG-2 graph if G[A] is an independent set, (χ(g) = B + 1). an NG-3 graph if G[A] is a 5-cycle, (χ(g) = B + 3). A K a, all B I a or C 5 K b none? I c C

30 NG-Graphs Recognition Algorithm Loop: Initialize k = 1 and partition V (G) as follows: A k = {v V (G) : deg(v) = k 1} B k = {v V (G) : deg(v) > k 1} C k = {v V (G) : deg(v) < k 1} Check if conditions (i) (v) hold. If all hold then check: If G[A k ] is complete, does A k + B k = k; If G[A k ] is an independent set, does B k + 1 = k; or If G[A k ] is a 5-cycle, does B k + 3 = k? If the answer is YES, then χ(g) = k and G is an NG-graph. If NO, add 1 to k. If k n, repeat the loop. If k = n + 1, then G is not an NG-graph.

31 NG-Graphs Subsequent Work Definition: A split graph is a graph whose vertex set can be partitioned into a clique and an independent set. Studied connections between NG-graphs and split graphs (Cheng/Collins/Trenk, 2016) Linear time recognition of NG-graphs using only the degree sequence.

32 NG-Graphs Current and Future Work Definition: An NGD-graph is a graph that satisfies χ D (G) + χ D (G) = n + D(G). Characterize the NGD-graphs. (Progress: characterization of the NG-graphs that are NGD-graphs) The distinguishing number of posets