Some Reflections on the Four Colour Conjecture
A NEW INDUCTION ARGUMENT WITH 4-COLOURATION PLUS  

The motif of this Blog is to pick up the pieces after the mistake in Blog 44, for which, further, profuse apologies. The heart of the problem is that this conjecture seems to cry out for an argument based on induction, and another possible induction argument can be based on the working hypothesis that any plane network can be 4-coloured in such a way that any (empty) five node circuit (composed of three triangles) can be 3-coloured, its node with four-links-to-others being the colour singleton. 

We start with the assumption that all fully triangulated, fully weeded networks up to n nodes can be 4-coloured, and also that any cluster of three adjacent triangles in such a network (see Diagram1) can have a pentagonal periphery composed of nodes which are colourable with only 3 colours, the node with four links being the colour singleton. (The effect of three colours on a five-node 

circuit will of course result in a 2,2,1 colour distribution. The ‘1’ is the colour singleton.) 

There are very large numbers of these ‘clusters of three adjacent triangles’. They may be called ‘3-clusters’. The principal question is: <<Can we deduce that if this assumption is valid for n=12 (which it is) and if it is valid for all fully triangulated, fully weeded networks of up to n nodes, then it is valid for any network of n+1 nodes (and therefore, by induction, for all values of n)?>>.

[An n-node network here will be called the ‘standard network’ and a typical n+1 node network will be the ‘extra network’.] 

We begin by considering such an ‘extra’ network of n+1 nodes, e.g. as shown in Diagram 2. This extra network is the focus of the argument.  We have to show that the working hypothesis applies to it.

We know from the reasoning of previous blogs in this sub-series that this extra network will contain large numbers of nodes of order 5, each surrounded with a pentagonal circuit of 5 nodes. 

Take any one of the order-5 nodes, X, and remove it, leaving a pentagonal hole and a total network of n nodes.  Now fill the pentagonal hole with two diagonals.         

(See Diagram 3) (This can be done in five different ways: each is a 3-cluster of three triangles.) The network is still standard, and both fully triangulated and fully weeded.

Applying the working assumption to it, we can deduce that this fully triangulated, fully weeded network of n nodes can be 4-coloured.  We can also deduce that

the nodes on the original circuit of X —now the circuit of a cluster of three triangles— can be coloured with a chosen set of three different colours.

Ergo, it is possible to remove the two inserted diagonals and replace them with the original node X coloured with the colour not included in the chosen set.  This in no way affects the coloration of the rest of the network.

So now the new, extra network of n+1 nodes has been shown to be 4-colourable.                                                          

It remains to show that the periphery of any 3-cluster in the extra network (i) can be coloured with only 3 colours, and (ii) that one of these nodes can be the colour singleton.

Fortunately most of the 3-clusters involved in the extra network are already

being assumed (from the working assumption) to be capable of being three coloured: with the node having the four links being the colour singleton.                   

What remains to be shown is that each of the new 3-clusters (present in the extra network but absent in the standard network) can be 3-coloured.  The colour singleton will, of course, be the one with four links inside the 3-cluster.      

Diagram 4 shows two examples of new 3-clusters hatched & dotted which were 

not present in the standard network.  They are of three kinds (a) entirely enclosed in the circuit of X (dotted), (b) composed of two triangles outside the circuit of X and one triangle inside (hatched), (c) composed of one triangle outside the circuit of X and two inside (not shown). 

Now consider the 3-clusters of type (a).An (a) type cluster like WVXYZ may be considered with the condition that node U (see Diagram 5) is the colour singleton which applies after removing X. It follows that nodes V, W, Y, Z can be coloured with only two colours.  So, taking the colour of X into account we have the 3-cluster WVXYZ with only three colours. The colour of X is different from each of the colours of V, W, Y, Z, so X can now be the colour singleton. 

This kind of argument applies to each of the five new 3-clusters of type (a). Thus

these ‘new’ dotted 3-clusters are covered. The working assumption here applies 

to the n+1 extra case.

{There are 5 new 3-clusters of type (a), 10 of type (b) and 10 of type (c) in Diagram 5.

Next, consider a typical new 3-cluster of type (b). Here we have a cluster of three triangles WPQVX, two of which are already present in the standard network. WPQVX is shown shaded/dotted in Diagram 6.  The third triangle of this new 3-cluster is shown dotted.

Now we can show that the pentagon WPQVZ is part of a standard network, so  the working hypothesis applies, and W and Q can have the same colour.  This means that P and X might have the same, or different, colours.

It has not been shown is that they must have the same colour, as required by the working hypothesis.  

However, if we start again with the extra networkand instead of removing X,

remove an order-5 node far away from X, the pentagon WPQVX becomes a normal 3-cluster within a standard network. 

This will suffice to establish that Q and X can have the same colours. A

similar argument applies to case (c). It also reinforces the argument given above

for the type (a) new 3-clusters.   

This putative proof will be revisited, reviewed and more fully explained in the April Blog (47).

CHRISTOPHER ORMELL 5^th March 2024 If you would like a free online copy of the P E R Narrative Maths Manifesto, send an email to per4group@gmail.com