Some Reflections on the new proof of the Four Colour Conjecture using induction

The new proof (See Blog 46) begins by posing the provisional hypothesis that all fully triangulated, fully weeded, networks of up to n nodes can be four coloured AND ALSO that this colouring can contain 3-coloured circuits of any or all of the 3-clusters within the n-node network.(A ‘3-cluster’ is a set of three adjacent triangles.)

The additional condition here (the ‘also’ above)  may look quite heavy, but of course, if the 4-colour conjecture is correct, the network consisting of order-5 nodes implanted in all the 3-cluster circuits of the initial n-node network, will also be capable of being 4-coloured. (See Diagram.) In which case all the circuits of all the “new” order-5 nodes will involve only three different colours.

The most striking feature of this new putative proof of the conjecture is its extreme simplicity.

It seems that, in the whirl of intense attention roused by the conjecture in the 19^th century, no one was minded to contemplate that an extra condition might be added to the conjecture. That the extra condition (the room for 3-colouring any or all of the circuits of the 3-clusters) was so substantial, was also unexpected.)

There will be cases where the removal of an order-5 node Z in the extra (n+1 node) network postulated in the new proof, will leave one of Zs former circuit nodes with only 4 links.  If so, what we get at this juncture is NOT a fully weeded network. However a weeding process can now take place. The first step, the initial removal of the order-4 node may lead to further order-4 nodes appearing. So these, too, can be weeded, and the weeding process might in theory lead to a complete deconstruction of the network. However whenever the weeding process is completed, it will contain m nodes where m < n. This leaves the number of nodes in the diminished network as less than n, and by the provisional hypothesis. 4-colourable.

The proof by induction method requires that the first appropriate configuration meets the condition set out by the provisional hypothesis. In this case the original fully triangulated, fully weeded network occurs when n=12 and all the nodes are of order 5.  (This may be called ‘the original network’.) It means that when one of the 3-cluster circuits of the original network is filled with an extra node, the singleton node of the original 3-cluster will drop down to order 4. The result of this is that it can be weeded by amalgamating opposite nodes —though clearly not the pair which had been previously linked together (before the two links of the original 3-cluster were removed). Now adding the node to the original network took it to n= 13, but weeding this node and amalgamating the two suitable opposite nodes reduces it to n= 11. This is now a network where 10 of the nodes are of order 5 and one is of order 4. So the order 4 node may now be weeded… which reduces the network to n=10… and so on. This is a long process which eventually leads to a simple triangle when n= 3. This is obviously 4-colourable, and by retracing the steps it leads to a valid 4-colouring of the original network. The same argument applies whichever 3-cluster of the original network is chosen. The original network has a great deal of built-in symmetry, which means that the same kind of process occurs whichever 3-cluster is targeted.  The detailed steps will be spelt out in the May Blog.

The simplicity of this new proof is extraordinary. To understand it better, it may help to bear in mind that the sheer multiplicity of possible 4-colourings is very large.   

The author’s interest in this problem was aroused when, at the age of 17 as a sixth former, he heard about the problem. His first partial insight was that one could replace the map regions by points and hence end-up with a network-colouring problem. During the intermediate 77 years he has managed to glean some further partial insights. In 1951 he did an undergraduate presentation in the (old) Mathematical Institute, Oxford on the problem. It was based on his re-discovery of the fact that any fully triangulated network contains umpteen Hamiltonian circuits.(These are circuits which embody all the nodes of the network in a single closed loop.) 

This meant that there were 2^n-2 initially viable ways to 4-colour the inside network created by the loop, and a similar number to 4-colour the outside network. To solve the problem it would be necessary to find 4-colourings which were common to both sets.    

In 1968 when he was a research Fellow at Leicester University he wrote a program in Algol which delivered 4-colourings of simple networks using Hamiltonian circuits based on the method outlined above. 

In 1999 he produced a booklet offering a semi-formal proof based on the idea of 4-colour whammies. (These were network patches which were evidently impossible to 4-colour. This informal proof was also based on induction. To establish it securely, though, a computer program was necessary to show beyond doubt that the simplest fully triangulated networks were whammy-free. But he was not able to find the time or the programming skills to establish the complicated software needed to look for such patches.)

The new proof by induction, as presented in blog instalment 46, arose directly from the disastrous failure of the putative proof offered in blog 45. Blog 48 will show in detail how the 4-colourability of the original network emerges from a sustained weeding process.

If you discover a glitch in the reasoning above, the author will welcome an email with details on per4group@gmail.com  He will  air an evaluation and response which will appear in the next blog (48).

CHRISTOPHER ORMELL 2^nd April 2024 If you would like a free online copy of the P E R Narrative Maths Manifesto, send an email to per4group@gmail.com