Infinite Problems and Finite Applications

1. Infinite Problemsand Finite Applications Nathaniel Dean Chair & Professor, Department of Mathematics Texas State University, San Marcos, TX 78666 nd17@txstate.edu The 28th Clemson Mini-Conference on Discrete Mathematics and Algorithms October 3, 2013

2. Some Infinite Problems • Chromatic Number of the plane • Integer & Rational Distances • Konig’s Infinity Lemma • Unit Distance Graphs • Single-walled Carbon Nanotubes (SWNTs) • Ulam’s Lattice Point Problem • Konig’s Infinity Lemma • Ulam Number • Incomprehensible Graphs

3. Graph G = (V, E)a set V of objects and a set E of (unordered) pairs of objects V = {1, 2, 3, 4, 5, 6, 7, 8, 9} E = {{1,2}, {1,3}, {1,7}, {1,3}, {2,6}, {3,4}, {3,6}, {4,6}, {4,7}, {4,8}, {5,6}, {5,7}, {5,9}, {7,8}, {7,9}, {8,9}}

4. edge link line vertex node point Graph G = (V, E)a set V of objects and a set E of (unordered) pairs of objects Edge List 12, 13, 17, 13, 26, 34, 36, 46, 47, 48, 56, 57, 59, 78, 79, 89 9 7 8 1 4 5 2 3 6

5. Points at Rational Distance What is the largest possible subset of points in the plane with no three of the points collinear and no four of the points concyclic, and having all interpoint distances rational? Jessica Poole – NREUP, Summer 2003 Tawfig Haj – M.S. Thesis, 2005 Charles Gabi – Undergrad Course, Fall 2006

6. Rational Distancesfor Points on a Circle (-7/10,12/5) (7/10,12/5) (-5/2,0) (5/2,0) (-7/10,-12/5) (7/10,-12/5)

7. Circles There exists an infinite set of points on the circle x2 + y2 = r2 where the distance between any pair of them is rational. Hence, there are arbitrarily many points, no 3 collinear, with integer distances.

8. Parabola How many points can be found on the (half) parabola y = x2, x ≥ 0, so that the distance between any pair of them is rational? > 4?

9. Points on a Parabola y y = x 2 d2 d3 d1 (r,r²) x

10. Infinitely Many Rational Distance Sets of Size 3 Proposition: There are infinitely many rational distance sets of 3 points on the parabola y = x2. Proof: Pick two rational numbers, say 1 and 3, and let those be the lengths of the segments AB and BC. Fix these distances and slide the points up the parabola. The distance AC will increase, bounded above by 4. Since the rationals are dense in the reals, there are many placements of the points where AC has rational length. Hence, there are infinitely many such triples.

11. More Recent Results • Avion Braveman (2001) and others produced three point solutions. • Campbell (1997) used elliptic curves to compute a four point solution and proved that there are infinitely many such sets. • Tawfig Haj (2005 Master’s thesis) by clever use of Pythagorean triples proved there are infinitely many 4-point solutions.

12. Ellipses?

13. Hyperbolas?

14. Konig’s Infinity Lemma Hierarchical Structure An infinite tree which is locally finite must contain an infinite path.

15. Konig’s Infinity Lemma  Proof Idea: Since there are finitely many branches, at least one of them must have an infinite subtree Go in that direction. 

16. Konig’s Infinity Lemma  Proof Idea: Find an infinite branch of the tree. Go in that direction.  

17. Konig’s Infinity Lemma  Proof Idea: Find an infinite branch of the tree. Go in that direction.   

18. Konig’s Infinity Lemma  Proof Idea: Find an infinite branch of the tree. Go in that direction.    

19. Proof Idea: Find an infinite branch of the tree. Go in that direction. Konig’s Infinity Lemma      

20. Application of KL to Coloring(applied later to Ulam Number) Corollary: Suppose G is a locally finite graph and any finite subgraph of G is k-colorable. Then G is k-colorable. Proof. Create a tree as follows. Enumerate the vertices of G as v1, v2, … , and let the nth level of the tree be given by the collection Ln of all k-colorings of v1, v2, … vn. Draw an edge from any k-coloring of v1, v2, … vnto a k-coloring of v1, v2, … vn+1 iffthe coloring of the n + 1 vertices extends our coloring of n vertices. This is a tree! It has infinitely many vertices, and by KL there's an infinite path. This path made of colorings that all agree with each other provides a k-coloring of all of G.

21. Chromatic Number of the Plane P Let G be the graph in the plane where V = R2 and two points are joined by an edge when their Euclidean distance is 1. Define P = (G) Hadwiger-Nelson (~1950): P = ? Any subgraph of G is called a unit distance graph. Unit Distance Graphs

22. Some Known UDGs Cycle Cn Petersen Graph Grid Graph

23. 4  P (The Moser Spindle)

24. P  7 (Hexagonal Tiling)

25. P  6 ?(Pritikin -1998) Every UDG of order 6197 is 6-colorable. Every UDG of order 12 is 4-colorable.

26. A UDG is maximal if the addition of any edge results in a graph which is not a UDG. Not Maximal Maximal

27. Number of Edges in a UDG • Purdy and Purdy (1988, CRAY computer) • Chilakamarri and Mahoney (1995, by hand) U(n) is the maximum size of a maximal UDG of order n. u(n) is the minimum size of a maximal UDG of order n.

28. Maximal UDGs on 7 vertices

29. The Minimal Forbidden Graphs on ≤ 7 Vertices

30. Separable UDGs All maximal UDGs are connected. There are no separable UDGs with < 8 vertices. There are only 3 separable maximal UDGs on 8 vertices.

31. ApproachGoal: Enumerate all maximal UDGs on 8 vertices • Find u(n) for 8 vertices • Show U(n) = 14 • Use maximal UDGs on 7 vertices to create a list of possible UDGs on 8 vertices. • Reduce the list of possibilities using minimal forbidden subgraphs. • Use mathematical algorithms to help identify UDGs from the remaining list. • Draw UDGs. • Find exact coordinates to verify UDGs.

32. Creating a List of Possible Maximal UDGs Starting with each of maximal UDGs on 7 vertices.

33. Process of Elimination Created list of possible non-isomorphic graphs from maximal UDGs on 7 vertices Eliminated graphs not 2-connected 1216 Eliminated graphs containing forbidden graphs Eliminated graphs without a triangle 118 Identified rigid subgraphsby hand Eliminated non-maximal UDGs 94 Proved or disproved each using Geometer’s Sketchpad Verified most results with algebraic geometry formulation Found exact coordinates from drawings and systems of equations

34. 61 Maximal UDGs on 8 Vertices

35. Ongoing Work Find all forbidden graphs on 8 vertices. Enumerate all maximal UDGs on 9 vertices. Proofs for general properties UDGs in higher dimensions Heawood graph

36. Carbon Nanotubes

37. Mathematical Programming Model of Bond Length and Angular Resolution of Carbon Nanotubes

38. Homomorphism A surjective map f: V(G)  V(H) of G onto H where • If xy E(G), then f(x)f(y)  E(H) or f(x) = f(y) and • If ab  E(H), then there exists x,y  V(G) such that f(x)=a, f(y)=b, and xy  E(G).

39. Homomorphism

40. Homomorphism

41. Another Homomorphism

42. Ulam Number (K. Chilakamarri and N. Dean, 1995) A homomorph H of G is a uniformly bounded homomorphif for some integer m every vertex x of H satisfies Ulam number u(G) = min {(H): H is a uniformly bounded homomorph of G}. • u(G)  (G). • H is a homomorph of G  u(H)  u(G). • F  G  u(F)  u(G).

43. Corollary: Ulam Number • Konig: Every infinite graph contains • infinitely many components, • a vertex of infinite degree, or • an infinite path. Corollary: Every finite homomorph of contains as a subgraph.

44. Drawing Infinite Graphs (K. Chilakamarri and N. Dean, 1995) A graph has a good drawing in k if it can be drawn in k so that •  a function h:++  every subset of k of diameter ≤ d contains ≤ h(d) verticesAND •  L  +  every edge has length ≤ L.

45. Drawing Infinite Graphs If G has a good drawing in a plane strip of fixed width, then u(G)  2. Shrinking each cell to a vertex yields a homomorph isomorphic to a collection of paths.

46. Drawing Infinite Graphs If G has a good drawing in the plane, then u(G)  6. Shrinking each cell to a vertex yields a homomorph isomorphic to a subgraph of the triangular grid.

47. Incomprehensible Drawing (not a good drawing) •  d  + such that, for any integer N, there is a region of diameter ≤ d containing ≥ N verticesOR • Edges are arbitrarily long. u(G)  3  every drawing of G in any strip [0,N] x R is incomprehensible. u(G)  7  every drawing of G in the plane is incomprehensible.

48. Final Remarks • Key IdeaKonig’s Infinity Lemma • Goals • Generalize it (infinite). • Apply it (finite).