*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. There are numerous ways of clubbing three adjacent triangles together. Any of these can have 3-colourable circuits as part of the 4-colouring of the whole network.)*

* Of course choosing a particular set of 3-clusters may end up with a few isolated triangles or triangle-pairs left (remaining) —unincorporated into further 3-clusters.*

*It is important to realise that in such cases ALL the 3-clusters identified can be, in principle. 3-coloured in addition to the 4-colouring of the whole network.*

* This follows, incidentally, from the fact that all the outside edges of the 3-clusters thus identified could be converted into the circuits of new order 5 nodes. (Each of these spaces could have its two diagonals removed and be replaced by a new node of order 5.) And since the enhanced network thus created is (we believe) 4-colourable, the key circuits involved will of course be 3-colourable. (The node which belongs to all three of the triangles will be the colour singularity.)*

* This blog focuses onto the ‘Starter Network’ composed of twelve nodes which is required to anchor the induction argument. It has to be shown in detail that this network is (a) 4-colourable and (b) that all the 3-clusters it contains, or could contain, are 3-colourable,*

*The Starter Network (see Figure 1) is composed of 12 nodes each of which is of order 5 and a total of 30 links. It is the smallest network therefore which is fully triangulated and also fully weeded (no nodes of order 3 or 4 are present). Its 4-colouring is shown in Figure 4 with the colours indicated by o @ X and | |. (The last of these is shown in Figure 4 as a box.)*

*It can also serve to show the kind of process by which a typical network can be 4-coloured. The steps are these:*

*One of the nodes (N) is removed (see Figure 2 with its hatched pentagon) and replaced by two new links which are the chosen diagonals of the pentagon formed by the circuit of N. The node where these two new diagonals meet (Y) will be the colour singularity. The opposite pair of nodes S, U will hopefully have the same colour and P, V will also have a different same colour. So the 3-cluster formed by PUYVS will be hopefully three coloured.**There are now two nodes in the remaining network of order 4 which can be weeded. Opposite nodes of the circuit rectangle can be amalgamated in each case, but care is needed here because it is possible to create colour contradictions. There is a need to ensure that the final circuit of the five nodes PMACS is 3-colourable. U can be removed by merging M & T or P & Q. On the other hand V (the other order-4 node) is more tricky. Merging Y and C works, but S and T doesn’t… because it creates four colours on the final circuit PMACS.**Figure 3 shows these amalgamations. Figure 4 shows the 4-colouration of the whole network.*

*This 12-node network contains a lot of symmetry, and similar operations can be conducted, starting with any one of the 11 remaining nodes. This is left as an exercise for the reader. *

*A prize of £1000 will be given to the first person to find a disabling glitch in this reasoning before Christmas 2024. Send your proposed glitch to **per4group@gmail.com** on or before 24*^{th}* Dec 3024, *

**CHRISTOPHER ORMELL 1 ^{st} May 2024 If you would like a free online copy of the P E R Narrative Maths Manifesto, send an email requesting this to to per4group@gmail.com**