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The (Degree, Diameter) Problem

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By Whitney Sherman

- 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

- 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).

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- 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

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- 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.

- 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!

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:

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

- 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

Table 1: Results for (3,3) planar 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.

Table 2: Summary of results for

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.

- 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.

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Moore Bound:

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- Can you create a planar (4,3) graph with n=16?
- How many edges must it have?
- What is the Moore Bound?