1. 1 A 4-Dimensional Graph Has at Least 9 Edges Math Colloquium 2015 December 2 Sonoma State University Roger House

2. 2 What we're going to talk about What is a graph? What is the dimension of a graph? What is the least number of edges a four- dimensional graph must have?

3. 3 What's a graph?

4. 4 Remember these graphs?

5. 5 And these?

6. 6 And graphs you can eat

7. 7 None of these graphs are the kinds of graphs we're interested in at the moment

8. 8 A different kind of graph We're going to deal with much different kinds of graphs, namely, mathematical graphs.

9. 9 Definition of a graph Definition: A graph G consists of a finite nonempty set V of vertices together with a set E of unordered pairs of distinct vertices of V. The pair e = {u,v} of vertices in E is called an edge of G.

10. 10 There will be a quiz on the definition But not on the one above On the one below

11. 11 A friendlier definition A graph is dots connected by lines. Dots and Lines by Richard J. Trudeau

12. 12 Fancy terminology The dots are called vertices. (Or points, or nodes, or...)‏ The lines are called edges. (The graph above has 7 vertices and 9 edges.)‏ Two vertices connected by an edge are adjacent. An edge is incident with its two vertices. The degree of a vertex is the number of edges incident with the vertex.

13. 13 Examples of graphs

14. 14 Yet more graphs

15. 15 Let's start at the beginning What is the simplest, most basic graph? The empty set? No. The definition says the set of vertices must be nonempty So this is it:

16. 16 What is the next simplest graph? How many vertices? Seems like 2 must be the answer So this is the next simplest graph:

17. 17 What is the third simplest graph? Three vertices? Are there any more graphs with 2 vertices? How about this one: Are there any more graphs with 2 vertices?

18. 18 Where do we go now? Three vertices

19. 19 More graphs with 3 vertices

20. 20 Wait a minute! Didn't we miss some?

21. 21 These two graphs are isomorphic iso-morphic = "same shape" = "equal form"

22. 22 The Graph Isomorphism Problem Is there an efficient algorithm for determining if two finite graphs are isomorphic? None is known But there is no proof that one does not exist NEWS BULLITEN 2015 Nov. 6: Laszlo Babai has reported that he has a quasipolynomial time algorithm for the graph isomorphism problem This is not an "efficient" algorithm, but it is a significant advance

23. 23 Academic commercial To learn more about isomorphism and the joys of abstract algebra: Math 320 Modern Algebra I Math 420 Modern Algebra II

24. 24 Graphs with V = 4 E = 0E = 1

25. 25 Graphs with V = 4 E = 2

26. 26 Graphs with V = 4 E = 3

27. 27 Graphs with V = 4 E = 3

28. 28 Graphs with V = 4 E = 4

29. 29 Graphs with V = 4 E = 5E = 6

30. 30 K n - complete graph on n vertices K n has n(n-1)/2 edges K 4 has 6 edges K 5 has 10 edges

31. 31 Complete graphs K 1 through K 6

32. 32 Find all graphs with 5 vertices Each dotted line is an edge or not an edge How many different configurations of edges are there?

33. 33 Find all graphs with 5 vertices K 5 has 5(5-1)/2 = 10 edges So there are 10 dotted lines, each one either an edge or not an edge So there are 2 10 = 1024 possible configurations So there are 1024 graphs with 5 vertices, right? No! Remember isomorphic graphs? There are 10 configurations with only one edge They're all isomorphic, so only 1 graph results

34. Are there enough graphs? V max E 2 E #graphs 1 0 1 1 2 1 2 2 3 3 8 4 4 6 64 11 5 10 1,024 34 6 15 32,768 156 7 21 2,097,152 1,044 8 28 268,435,456 12,346 9 36 68,719,476,736 274,668 10 45 35,184,372,088,832 12,005,168

35. Consider these graphs... K 1,1 K 1,2

36. And these... K 1,3 K 2,2

37. What's the structure? K 2,3 K 3,3

38. 38 Complete bipartite graphs If the set of all vertices can be partitioned into sets V and W such that every edge connects a vertex in V to a vertex in W, then the graph is called bipartite If a bipartite graph has as many edges as possible, then it is called a complete bipartite graph Notation: K m,n, where m = |V| and n = |W| How many edges in K m,n ? |K m,n | = mn

39. 39 Not all bipartite graphs are complete

40. Cyclic graphs C3C3 C4C4 C5C5 C 10 C8C8 C6C6

41. Trees A tree is a connected graph with no cycles What if a single edge is added to a tree? For a tree: |V| = |E| + 1 In general: |E| ³ |V| - 1

42. 42 What now? This concludes a crash introduction to graph theory We move on to our next main question:

43. 43 What is the dimension of a graph?

44. 44 The dimension of a graph In 1965, the illustrious trio P. Erdős, F. Harary, and W.T. Tutte published a paper entitled On the dimenion of a graph The first thing they did in this paper was to define the dimension of a graph

45. 45 The dimension of a graph The dimension of a graph G, denoted dim(G), is the minimum n such that G has a unit- distance representation in  n, i.e., every edge is of length 1. The vertices of G are mapped to distinct points of  n, but edges may cross Intuitively: Given a graph, construct a model where every edge is of length 1, and then figure out whether it can live in 1-space, 2- space, 3-space, 4-space,..., n-space

46. What is the dimension of K 1,1 ? dim(K 1,1 ) = 1

47. What is the dimension of K 1,2 ? dim(K 1,2 ) = 1

48. What is the dimension of K 1,3 ? dim(K 1,3 ) = 2

49. What is the dimension of K 2,2 ? dim(K 2,2 ) = 2

50. What is the dimension of K 2,3 ? 1 2 3 A B dim(K 2,3 ) = ?

51. What is the dimension of K 2,3 ? 1 2 3 A 1 2 3 B

52. 1 3 A 1 2 3 B

53. 2A 1 2 3 B dim(K 2,3 ) > 2

54. 54 K 2,3 in three dimensions A B x y z 2 3 1 A, B to origin: √3/2 circle: radius = 1/2 dim(K 2,3 ) = 3

55. What about K n,m for n,m ≥ 3? There are a lot of points on that circle dim(K 2,m ) = 3 for m > 3 What is dim(K 3,3 )? What's a lower bound on dim(K 3,3 )? Since K 2,3 is a subgraph of K 3,3 dim(K 2,3 ) £ dim(K 3,3 ) So dim(K 3,3 ) ≥ 3

56. A lower bound for dim(G) Thm: If H is a subgraph of G, dim(H) £ dim(G) Proof: Say dim(G) = n Consider an embedding of G in  n Remove vertices and edges of G so that only H is left H is embedded in  n so dim(H) £ n 

57. What is the dimension of K 3,3 ? 2 A C 1 3 B dim(K 3,3 ) = ?

58. What is the dimension of K 3,3 ? We know dim(K 3,3 )  3 Might dim(K 3,3 ) = 3? Consider three spheres S A, S B and S c of radius 1 centered at A, B, and C, respectively S A and S B intersect in a circle on which vertices 1, 2, and 3 lie This circle must also lie on S C This can only happen if S C is one of S A or S B Which means C = A or C = B ⇒ Ü

59. What is the dimension of K 3,3 ? So dim(K 3,3 )  4 In fact, dim(K 3,3 ) = 4 To show this, we consider two circles of radius 1/Ö2 in  4 : C 1 : x 2 + y 2 = ½ C 2 : z 2 + w 2 = ½

60. What is the dimension of K 3,3 ? Pick any point P = (x, y, 0, 0) on circle C 1 Pick any point Q = (0, 0, z, w) on circle C 2 The distance between P and Q is the square root of (x-0) 2 + (y-0) 2 + (0-z) 2 + (0-w) 2 = x 2 + y 2 + z 2 + w 2 = (x 2 + y 2 ) + (z 2 + w 2 ) = ½ + ½ = 1

61. What is the dimension of K 3,3 ? So every point P on circle C 1 is at a distance 1 from every point Q on circle C 2 Pick any three distinct points on C 1 and call them A, B, and C Pick any three distinct points on C 2 and call them 1, 2, and 3 Insert an edge of length 1 from each letter to each digit We have an embedding of K 3,3 in  4 

62. What is the dimension of K m,n ? What about dim(K m,n ) for m,n ≥ 3? There are a lot of points on those two circles dim(K m,n ) = 4 for m,n ≥ 3

63. What is the dimension of K n ? K 3 can be represented as an equilateral triangle: dim(K 3 ) = 2 K 4 can be represented as a regular tetrahedron dim(K 4 ) = 3 K 5 can be represented as...? dim(K 5 ) = ? An exercise for the perspicacious student: Show that dim(K n ) = n - 1

64. Basic results about dimension dim(K n ) = n - 1 dim(K n - e) = n - 2 dim(K 1,1 ) = dim(K 1,2 ) = 1, dim(K 1,m ) = 2 for m³3 dim(K 2,2 ) = 2, dim(K 2,m ) = 3 for m ³ 3 dim(K m,n ) = 4 for m, n ³ 3 dim(C n ) = 2 for C n a cyclic graph of order n ³ 3 dim(tree) £ 2 if H is a subgraph of G then dim(H) £ dim(G)

65. Questions Now we have answered our first two questions: What is a graph? What is the dimension of a graph? Only one question is left:

66. 66 What is the least number of edges a four-dimensional graph must have?

67. Exactly what is the question? In 2009 The Mathematical Coloring Book by Alexander Soifer was published In this book a question posed by Paul Erdős in 1991 appears What is the smallest number of edges in a graph G if dim(G) = 4? In the rest of this talk we will answer this question

68. Where to start? We have seen two graphs of dimension 4 K 5 : V=5, E=10 K 3,3 : V=6, E=9

69. Minimum number of vertices? Since dim(K n ) = n-1, we know dim(K 5 ) = 4 Since dim(K n - e) = n-2, we know dim(K 5 - e) = 3, so every proper subgraph of K 5 has dimension at most 3 So a four dimensional graph with a minimum number of edges must have more than 5 vertices Therefore we need to look at graphs with 6 or more vertices

70. Number of edges? Remember that a connected graph must have at least one more vertex than it has edges, i.e., |E| ³ |V| - 1 So, if |V| = 6, it must be that |E| ³ 6 - 1 = 5 Therefore we need to look at graphs with 5 or more edges We already have a four-dimensional graph with 9 edges (K 3,3 ), so we need not consider graphs with more than 9 edges

71. Maximum number of vertices? Using |E| ³ |V| - 1 again with |E| = 9, we have 9 ³ |V| - 1, so |V| £ 10 Therefore we need to look at graphs with 10 or fewer vertices To sum up: Find a four-dimensional graph with 6 £ |V| £ 10 5 £ |E| £ 9

72. Fill in the blanks # edge #vert 98765 6 7 8 9 10

73. |V| = |E| + 1 is easy: A tree # edge #vert 98765 6tree 7 8 9 10tree

74. dim(tree) £ 2 We return to the question: What if a single edge is added to a tree?

75. Add one edge, get one cycle

76. |V| = |E| is easy: A cycle # edge #vert 98765 6cycletree 7cycletree 8cycletree 9cycletree 10tree

77. Add two edges and get what?

78. Two cycles with no common edge

79. Two cycles with edges in common

80. Reduce to essentials

81. Make all edges have unit length

82. 3-routes When two edges are added to a tree so that the result is two cycles sharing at least one common edge, the resulting graph is called a 3-route A 3-route consists of three paths which have nothing in common except their end vertices So there is a left path, a middle path, and a right path from one end vertex to the other end vertex

83. Another example of a 3-route u v Another exercise for the perspicacious student: Show that dim(3-route) = 2 with one exception.

84. |V| = |E| - 1 is easy: A 3-route # edge #vert 98765 63-routecycletree 73-routecycletree 83-routecycletree 9cycletree 10tree

85. We're getting closer and closer Now only these three cases are left: 6 vertices, 8 edges 6 vertices, 9 edges 7 vertices, 9 edges There are 156 graphs with 6 vertices and 1044 graphs with 7 vertices Way too many To the rescue: An Atlas of Graphs by R.C. Read and R.J. Wilson

86. 6 vertices, 8 edges

87. Narrowing down The Atlas lets us pare down to fewer graphs: 6 vertices, 8 edges: 24 graphs 6 vertices, 9 edges: 21 graphs 7 vertices, 9 edges: 131 graphs Total: 176 graphs It's still too many But let's take a look at that Atlas page again

88. 6 vertices, 8 edges

89. Cut Vertex

93. Narrowing down yet more These graphs are not of interest: A graph with vertices of degree 0 or 1 A graph which is not connected A graph which has a cut vertex Yet another exercise for the perspicacious student: If G is a four-dimensional graph with a minimum number of edges, then G cannot contain a cut vertex Terminology: A graph with no cut vertex is said to have vertex connectivity ³ 2

95. All the blanks are filled in # edge #vert 98765 61493-routecycletree 7203-routecycletree 83-routecycletree 9cycletree 10tree

96. Narrowed down enough? So we are now down to this many graphs: 6 vertices, 8 edges: 9 graphs 6 vertices, 9 edges: 14 graphs 7 vertices, 9 edges: 20 graphs Total: 43 graphs Lacking a brilliant flash of insight, we look at all of the 43 graphs and see if we can embed them in 2-, 3-, or 4-dimensions

97. G147 (V=6, E=8)

98. G580 (V=7, E=9)

99. G146 (V=6, E=8)

100. G171 (V=6, E=9)

101. Only 39 more to go Fortunately, we are running out of time, so you won't have to look at the details of the 39 remaining embeddings Here is the breakdown by dimension: 2-dimensional: 27 graphs 3-dimensional: 15 graphs 4-dimensional: 1 graph Total: 43 graphs Here are all 43 embeddings:

102. 2-dimensional - part 1

103. 2-dimensional - part 2

104. 3-dimensional - part 1

105. 3-dimensional - part 2

106. 3-dimensional - part 3

107. There's only one left Of the 43 candidate graphs, 42 are 2- or 3- dimensional, leaving just one graph, which answers the original question A 4-dimensional graph must have at least 9 edges, and there is only one 4- dimensional graph with 9 edges: K 3,3 I had a lovely unit-distance drawing of K 3,3 in 4- dimensions, but I lost it. This will have to do:

108. 4-dimensional graph with 9 edges

109. That's all, folks For all the details see: rogerfhouse.com Thank you