Posts Tagged ‘ cage ’

Thesis Draft 5

From here on, each draft of my thesis will be constructed using PCTeX.

Thesis Draft 5

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Thesis Draft 4 (PCTeX)

Dr. Sharma asked me to rewrite what I had so far for my thesis using PCTeX. PCTeX is a software that uses TeX, a mathematical language, to produce documents. I have been learning the software and language as I go, so these drafts are truly works in progress.

Thesis Draft 4 (PCTeX)

Thesis Draft 3

This draft reflects the new organization of my thesis and main topics we aim to cover. Some of the topics have already been discussed, but I did not have time to add them to this draft of the paper. See blog posts for up-to-date information on what topics I have already researched.

Thesis Draft 3

Thesis Draft 2: Examples and LCF Notation

After meeting with my adviser, he suggested some changes for my terminology section and added some new sections. I have now also included personal examples of cubic graphs that I have drawn and LCF notation.

Thesis Draft 2: Examples and LCF Notation

Thesis Draft 1: Terminology and Known Cubic Cages

I have begun to type up the beginning stages of my thesis, even though we have not completely defined it’s format. I know it will be focussed on the study of cubic graphs and their applications, so my adviser asked me to type up the terminology and known cubic cages sections to start with. I will be including drafts frequently as we add and change material.

Thesis Draft 1: Terminology and Known Cages

3-6 Cage (corrected)

Tonight, as I was pulling apart my 3-6 cage and attempting to make it into a Hamiltonian cubic graph, I realized it was in fact not a 3-6 cage. A cycle of length 5 appeared that had not been obvious to me before (highlighted below).

5-3 graph on 14 vertices

After I had constructed what I thought was a 6-3 Cage, I looked it up online to compare the common construction to mine. Because of that, I already know how to construct the 6-3 Cage, so I included it below.

6-3 Cage

3-6 Cage (false)

After lots of erasing and retries, I think I finally created a 3-6 Cage. The basis for my graph was two cycles of length 6, one in the shape of a hexagon and contained in that one was a twisted hexagon (4 edges cross, but do not have a joining vertex in the middle). From here, it was a process of elimination to figure out where the remaining edges could be placed. I will be presenting my graph to my adviser tomorrow to see if it is correct, but from the checks I’ve done, the girth is 6 and there are 14 vertices…which is the amount he told me the 6-3 cage is on.

cubic graph with girth 6 on 14 vertices

********** Later proved false, see this post ***********

 

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