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Showing posts from June, 2013

Tutte's Twist Operation on Cubic Graphs

There is an interesting book by W. T. Tutte called 'Graph Theory as I have known it' which is a cross between a normal mathematical text and a biography. So it's a description of the areas he was interested in, and his theorems. One thing that interested me was the use of a 'twist' operation on cubic graphs like so:

Where for the edge between vertices x and y labelled 'A' we reconnect the surrounding edges to form the arrangement on the right hand side. So detach edge D from y and connect it to x, and vice versa with edge C. The lower part of the picture shows what happens for a loop-edge - it transforms to a multi-edge.

This operation is used on a family of 'base' graphs looking like this:

with the first in the list is a vertexless loop graph - that is, it has no vertices and a single edge. From these base graphs, the twist operation can form any cubic graph. Note that all of Un are cubic with 2n vertices.

For example, from U3 we can get to both of…