Loop Edge Estimation in 4-Regular Hamiltonian Graphs
In knot theory, a knot is defined as a closed, non-self-intersecting curve embedded in three-dimensional space that cannot be untangled to produce a simple planar loop. A mathematical knot is essentially a conventional knot tied with rope where the ends of the rope have been glued together. One way to sample large knots is based on choosing a 4-regular Hamiltonian planar graph. A method for generating rooted 4-regular Hamiltonian planar graphs with n vertices is discussed in this thesis. In the generation process of these graphs, some vertices are introduced that can be easily eliminated from the resulting knot diagram. The main result of this thesis is the estimation of the expected number of loop edges in a 4-regular Hamiltonian planar graphs of n vertices; in particular, it is shown that the expected number of loop edges L(n) in such a graph has asymptotic order n/6.
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