The (Degree, Diameter) ProblemBy Whitney Sherman

Land of Many PondsThere 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 doesnt eat the duck.

DuckDragonMediator

VocabularyDegree 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).

ExampleAll 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) graphGDiameter Tablelabcdefghijk

Real World ApplicationIn 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.

Moore BoundFor 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!

Hilbigs Theorem

Both of the exceptions in this theorem are non-planarThis theorem can be used to find planar (3,3) graphs whenExcept 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.Peterson Graph

Construction of (3,3)Start with the Hamiltonian cycle on n verticesAdd 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 constructedIn 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.

Pratts Results using Hilbigs TheoremTable 1: Results for (3,3) planar graphs.

Vertices (n)1-factors of Cases Diameter 3PlanarIsomorphism Classes43100061540008105311818310945293268906121039533261580242

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

Final Results Using Hilbig, McKay, and RoyleTable 2: Summary of results for

Vertices (n)Connected Cubic GraphsHamiltonian Diameter 3Planar (3,3) graphs468101214161820 1251985509406041301510489 1251780474483139635495991 0031534341411 003620000

Further ResearchThis problem continues to be researched on larger graphs.In turn, new theorems are brought about.Zhangs 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.

It all comes togetherThe 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.

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