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Published online by Cambridge University Press: 13 March 2010
Prologue
Now Albert had heard about Lions,
How they was ferocious and wild -
To see Wallace lying so peaceful,
Well, it didn't seem right to the child.
So straightway the brave little feller,
Not showing a morsel of fear,
Took his stick with its ‘orse's ‘ead ‘Andle
And pushed it in Wallace's ear.
You could see that the Lion didn't like it,
For giving a kind of a roll,
He pulled Albert inside the cage with ‘im,
And swallowed the little lad ‘ole.
(From “The lion and Albert” by Marriott Edgar (1932))
Cubic graphs are a class of graphs which have attracted considerable interest, not least because of the intimate relationship between planar cubic graphs and the Four Colour Theorem. One particular field of interest has always been the symmetries of cubic graphs. Another has been the existence of cubic graphs with specified constraints on their diameter or girth. These two fields of interest coincide in the study of cages i.e. the smallest possible cubic graphs with given constraints on their girth. In a certain sense some cages turn out to be the most symmetric graphs of all. Of course, P is a cage.
Cages
A regular graph of degree 1 has no cycles. A regular graph of degree 2 has arbitrary girth. So we define a graph which is regular of degree r and of girth g to be an (r, g)-graph only for r ≥ 3. Clearly g is also greater than or equal to 3. By the arguments of Section 1.5 the Petersen graph is a (3, 5)-graph.
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