The (Degree, Diameter) Problem

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. 1 3 3 3 1 3 2 2 2 0 2 3 2 1 2 2 1 2 3 0 3 1 1 2 2 3 2 0 1 3 3 2 2 3 2 2 2 2 1 2 2 1 2 1 3 2 3 a 2 1 1 0 1 3 0 3 3 e 2 2 0 2 1 1 f 3 3 3 2 3 2 1 0 b 1 2 3 1 2 2 2 1 2 3 3 3 2 1 3 0 2 1 1 1 0 2 2 2 2 1 3 2 3 2 2 1 c 3 3 3 g 0 3 2 2 3 2 3 2 1 2 1 1 2 3 1 1 2 1 3 2 d h 2 2 0 3 e a 1 i j 2 1 k 0 l b f 2 c g 3 i d h 3 j 2 k 3 3 1 2 1 2 1 l Example • All 12 vertices of G are of degree 3, so G is 3-regular. • The diameter table shows the distances between each vertex. Diameter Table G G is a planar (3,3) graph d k e j c a l f h i b g

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 • 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: • For example: The Moore bound on a 3-regular, non-planar graph with 20 vertices and a diameter of 3, is 22 A (3,3) Non-planar graph on 20 vertices (largest known) Note: The Moore Bound is not necessarily achieved!

7. 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: Hilbigs Theorem 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. • Both of the exceptions in this theorem are non-planar • This theorem can be used to find planar (3,3) graphs when Peterson Graph

8. Construction of (3,3) • 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. • 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

9. Pratt’s Results using Hilbigs Theorem Table 1: Results for (3,3) planar graphs.

10. Examples of Planar (3,3) nth Order Graphs Haewood graph • 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 • 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.

11. Final Results Using Hilbig, McKay, and Royle Table 2: Summary of results for

12. Further Research Zhang’s Theorem (1985) • This problem continues to be researched on larger graphs. • In turn, new theorems are brought about. • 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. a b a 3 0 3 2 m 1 e h i b f 1 3 0 3 n 3 edges g 3 4 2 o 1 c 2 o f 1 i h 2 g d 2 2 1 p j j 1 a 1 k n p e k Moore Bound: m l b c d 3 l Class Example • Can you create a planar (4,3) graph with n=16? • How many edges must it have? • What is the Moore Bound?