1. The (Degree, Diameter) Problem By Whitney Sherman

2. Land of Many Ponds There exists a mystical place call it the Land of Many Ponds. Three things live there, a duck, a dragon, and a ‘mediator.’ The duck can move only to 1 pond at a time. The dragon can move 2 and the ‘mediator’ 3. The dragon decides to try and find the duck. It is up to the mediator to get to the duck at the same time as the dragon does so he doesn’t eat the duck. Duck Dragon Mediator

3. Vocabulary Degree is the number of edges emanating from a given vertex. A graph is called regular if all of the vertices have the same degree. The distance from one vertex x to another vertex y is the smallest number of moves that it takes to get there. The diameter of a graph is the longest distance you can find between two vertices. So the diameter of a graph is the maximum of the minimum distances between all pairs of vertices. A given graph G is has Degree, and diameter and this is expressed as ( where is the maximum degree over all the vertices).

4. Example All 12 vertices of G are of degree 3, so G is 3-regular. The diameter table shows the distances between each vertex. G is a planar (3,3) graph G Diameter Table 12102123323 01211222333 10122331223 21013232213 21231320122 12320211232 23212013322 23321102231 33332212120 32232321011 32123232102 bcdafghejkil 2 3 3 1 2 1 2 1 e f g h a b c d i j k l 2 3 3 332231232110 l a b c d e f g h i j k

5. Real World Application In designing large interconnections of networks, there is usually a need for each pair of nodes to communicate or to exchange data efficiently, and it is impractical to directly connect each pair of nodes. The problem of designing networks concerned with two constraints: (1) The limitation of the number of connections attached to every node, the degree of a node, and (2) The limitation of the number of intermediate nodes on the communication route between any two given nodes, the diameter. Consequently the problem becomes the degree/ diameter problem So the goal is to find large order graphs with small values.

6. Moore Bound For example: The Moore bound on a 3-regular, non-planar graph with 20 vertices and a diameter of 3, is 22 The order (i.e. the number of vertices) of a graph with degree where is > 2 and with diameter is bounded by the Moore Bound. The Moore bound is found by this equation: A (3,3) Non-planar graph on 20 vertices (largest known) Note: The Moore Bound is not necessarily achieved!

7. Hilbigs Theorem Both of the exceptions in this theorem are non-planar This theorem can be used to find planar (3,3) graphs when Except for the Peterson graph and the graph obtained from it (by expanding one vertex to a triangle), every 2-connected, d-regular graph on at most vertices is Hamiltonian. A graph G is said to be k- connected if there does not exist a set of k-1 vertices whose removal disconnects the graph Example of 2-connected graphs: Peterson Graph

8. Construction of (3,3) Start with the Hamiltonian cycle on n vertices Add to it, a 1-factor (Recall: A 1-factor is a perfect matching in a graph i.e. spanning subgraph which is 1-regular ) of The number of 1-factors of (n even) is given by: However, we are not interested in those 1-factors that contain an edge of the Hamiltonian cycle because they would give us a multigraph. So we consider every 1-factor of - where translates to “a 2-factor.” This gives a simple cubic graph and by Hilbigs theorem any (3,3) graph on at most 12 vertices can be constructed In any attempt to draw these graphs recall the first theorem of graph theory: that the sum of all the degrees of all the vertices is twice the number of edges. So say you attempted to make a (3,3) graph on 12 vertices… you know that the graph has to have 18 edges.

9. Pratt’s Results using Hilbigs Theorem Vertices (n)1-factors ofCasesDiameter 3PlanarIsomorphism Classes 431000 6154000 81053118 3 10945293268906 121039533261580242 Table 1: Results for (3,3) planar graphs.

10. Examples of Planar (3,3) nth Order Graphs n=8 Recall Table 1: there are 3 graphs that have these properties. n=10 Recall Table 1: there are 6 graphs that have these properties n=12 Recall Table 1: There are 2 graphs that have these properties. n=14 There are 509 connected cubic graphs on n=14. Only 34 with a diameter of 3, and none are planar. n=16 There are 4060 connected cubic graphs on n=16 Only 14 have diameter 3 and none are planar. n=18 There are 41301 connected cubic graphs on 18 vertices 1 has diameter 3 but it is not planar Haewood graph

11. Final Results Using Hilbig, McKay, and Royle Table 2: Summary of results for Vertices (n)Connected Cubic Graphs HamiltonianDiameter 3Planar (3,3) graphs 4 6 8 10 12 14 16 18 20 1 2 5 19 85 509 4060 41301 510489 1 2 5 17 80 474 4831 39635 495991 0 3 15 34 14 1 003620000003620000

12. Further Research This problem continues to be researched on larger graphs. In turn, new theorems are brought about. Zhang’s Theorem (1985) Every 4-regular graph contains a 3-regular sub graph. Using this theorem, one can find planar graphs on a fixed number of vertices n, by adding 1-factors to the planar graphs on n vertices for all with (since adding edges does not increase the diameter) and (K is the connectivity, if K is unknown, K=1). Peterson (1891) A graph is 2-factorable it is regular of even degree. A 2-factorization of a graph is a decomposition of all the edges of the graph into 2-factors i.e. a spanning graph that is 2-regular Hartsfield & Ringel Theorem (1994) Every regular graph of even degree is bridgeless. This shows that when is even, a connected regular graph is 2- edge-connected.

13. It all comes together… The pond example came about because “the land of many ponds” is a (3,3) planar graph on 12 vertices. I was interested to find if there was a graph of larger order that still held these properties. As it turns out there is not, Pratt proved this in 1996.

14. Class Example Can you create a planar (4,3) graph with n=16? How many edges must it have? What is the Moore Bound? ba e f g h a b c d i j k l m n o p 3 3 3 2 0 1 2 2 1 1 1 3 3 3 2 2 2 3 4 1 0 1 1 e f g h a b c d i j k l m n o p edges Moore Bound: