Stonehill College

Data Structures and Discrete Mathematics Learning Community

Professors Ralph Bravaco and Shai Simonson


Introduction

This lab will explore an algorithm to 4-color a planar graph.  The algorithm is based directly on a proof.  Your ability to code the algorithm will be completely dependent on whether or not you understand the proof.  This is a nice example of mathematics (even proofs) motivating computer science.  Furthermore, it will be a tangible test of whether you really can understand a proof.  

The lab also explores the notion of theory versus experiment in computer science.  The proof we discuss is a famous one with a flaw!  Therefore, your algorithm will theoretically not run correctly.  Nevertheless, experiments with this pseudo-correct algorithm show that it runs correctly much of the time.  There is no theory powerful enough to predict when it will run correctly and when it will not, short of actually trying to run the algorithm.  This illustrates the subtle relationship between theory and practice (experiment) in computer science

The Four-Color Theorem


The four color theorem states that any planar map can be colored with at most four colors.  Below is a map of the United State colored using four colors  


                                                             US Map colored with 4 colors

In graph terminology, this means that using at most four colors, any planar graph ( a graph that can be drawn without any of its edges crossing) can have its nodes colored such that no two adjacent nodes have the same color.   The map version of the theorem uses faces in place of nodes, where any two faces are adjacent if and only if they share a border (an edge).  Note that a single point like the four corner border between Utah, Arizona, Colorado, and New Mexico is not considered a border for our purposes.

Make sure you understand that any map can be considered a graph where the faces of the map becomes nodes of the graph.  For example, the four states Utah, Arizona, Colorado, and New Mexico considered alone as a graph would look like this: (insert simple square).  We will henceforth think of a map as a graph.

History

The four-color theorem was conjectured in 1852 and proved in 1976 by Wolfgang Haken and Kenneth Appel at the University of Illinois with the aid of a computer program that was thousands of lines long and took over 1200 hours to run. Since that time, a collective effort by interested mathematicians has been under way to check the program. So far the only errors that have been found are minor and were easily fixed. Many mathematicians accept the theorem as true.  There are many sites that summarize the history and details of the problem.   Here is a good example.

Perhaps the most interesting feature of the 1976 proof is that it made use of a computer program to examine thousands of cases.  This was at the time, and still is to some extent, a controversial and unprecedented use of computers in the actual proof of a theorem.  The computer acted not merely as a tool for exploration, or as a way to implement a theory, but instead as a fundamental part of the fabric of the proof.  Although the mathematics community accepted the correctness of the proof, there were and are still are those who question the proof's validity on philosophical grounds.  "What makes a proof a proof?"  they ask.  Practically speaking, it is the confidence that competent mathematicians have checked the logic and meticulously scrutinized the details.  But in this proof, the collective effort of hundreds of mathematicians for dozens of years would be necessary to check each case that the computer checked in a few hours.  Even if such an effort were possible, it is hardly a feeling of confidence to know that any small error by anyone of hundreds of people would render the proof incorrect.  If it is not practical then to cut the computer out of the picture, then how do we know for sure that the computer program is correct.  Perhaps we could get mathematicians to formally check the correctness of the program, but how would we know for sure that the program ran correctly with certainty.  It would not do to say that we are 99.999% sure the proof is correct.  In math, a correct proof means 100%.   These are some of the issues people have with this proof.  Still the proof is accepted by the mathematical community at large, although even those who have no objection to its correctness, long for a more human elegant proof.

The site referred to above has this to say:

"The proof of the 4-Color-Theorem is a c and computing machines in mathematics. Is it fair to accept as true what a computer can verify, even though no single person can? Does the nature or texture of what humans can discover about their world change with the use of computers as thinking tools? Computers are huge and powerful, but finite; will using them as thinking tools be ultimately limiting? These issues are raised and considered in Ad Infinitum: The Ghost in Turing's Machine by Brian Rotman, and Pi in the Sky by John Barrow."

Ironically, a very elegant "proof" of the four color theorem was "published" by Kempe in 1879.  The proof underwent the usual scrutiny of experts and was published in a respectable journal.  Eleven  years later Heawood published a paper pointing out a flaw!  This rare occurrence of a refereed journal article with a  proof containing a serious logical error foreshadowed the even stranger results of the 1975 computer aided proof.  It happens often enough that a journal referee will find errors in proofs, but often these errors are easily corrected by the author.  The rare times when the errors are so serious that the author is unable to correct them are almost always caught before the paper goes to press.  It was not so with Kempe's proof.  You may be wondering what kind of clever subtle unfixable flaw lurked in Kempe's proof?  You will soon find out.  Perhaps you will find the flaw yourself before it is pointed out explcitly. 

Luckily, Heawood was able to salvage some good results from the remains of Kempe's work.  A number of excellent techniques and a five-color theorem remained.

In this lab, we study Kempe's proof and do some programming experiments to explore his methods, and see if they can be used as a (probabilistic or approximation) coloring algorithm.  A probabilistic algorithm is one that gets the correct aswer most of the time;  an approximation algorithm is one that gets close to the correct answer every time.  Irony is again the theme here, as the algorithms based on Kempe's theory will work in practice but defy a rigorous analysis.  We have a proof that is accomplished only with the help of a computer, and an algorithm that seems to work even though it is based on a wrong proof!  This is truly a case study of the relationship between computer science and mathematics.

Kempe's "Proof" by Induction of the Four Color Theorem

Kempe's proof is constructive in the sense that the details of the proof imply a recursive algorithm for coloring.  Kempe's proof is by indcution on the number of regions in the map (or nodes in the graph).   Like many difficult inductive proofs, he uses strong indcution.  This means he attempts to show that a graph with n nodes can be four-colored and assumes that all graphs with fewer than n nodes can be four-colored.  This style is in mild contrast to the simple inductive proofs that we use for formulae like: 1+ 2+ 3 + ... + n = n(n+1)/2 where we start with nth case and work toward the n+1st case.

Before he gets to the inductive idea, Kempe begins with a clever way to categorize the possible graphs.  He first states that we might as well consider graphs all of whose faces are triangles.  These maximal planar graphs include the maximum number of edges possible, and if they can be four-colored, then so can other graphs with fewer edges. He then proves a lemma (a helper theorem) that any maximal planar graph must have at least one node with degree 5 or less.  The proof uses Euler's Theorem for planar graphs and a counting argument to show that if all the nodes had degree 6 or higher, then the graph would not be planar.  The details can be found below but they are not relevant to the main thread of the proof, and can be read independently.

If we denote the number of nodes, edges, and faces (i.e., the bounded regions) of a planar graph by V, E, and F respectively, then Euler's theorem for a plane (or a sphere) is V - E + F = 2 (the outside face counts too).  Each face of a maximal planar graph is bounded by three edges, and each edge is on the boundary of two faces, so we have F = 2E/3.  Euler's formula for a complete planar graph then becomes simply E = 3V - 6.  The degree of a node is the number of edges incident to it.  Now each edge is connected to two nodes, so the sum of the degrees of all the nodes 2E = 6V - 12, and hence the average degree per node is 6 - 12/V < 6,  (assuming the graph is finite).  This implies that at least one vertex has degree five or less.

Exercise 1:  Prove Euler's theorem (thereby completing the proof above that in a planar graph at least one vertex has degree five or less).  Your proof should use strong induction on the number of faces in the graph.  You can find this proof in many places.  If you can not figure it out on your own, then at least write it up in your own words, showing clearly that you understand it!

Kempe concentrates on the node with degree five or less, and enumerates four cases:  degree = 2, 3, 4, 5 (the degree can't equal one because then there would be a non-triangular face in the graph).

Degree = 2

If there is a node x with degree two, then remove it and all edges incident to it.  By induction, the graph that remains is four-colorable.  Let the colors of the two nodes that were incident to x be B and G.  Now add the node x back in, and color it R or Y.  This is a four-coloring of the original graph.

Degree = 3

Similar to the previous case.  This time, color x differently from the colors on the three nodes incident to it.  Hence if the three nodes are colored B, G and R, then color x with Y.

Degree = 4

This is the first difficult case, and the place where Kempe needed to invent the idea of what is now called a Kempe chain.  In this case, there are four nodes incident to x.  As before, remove x, and color the remaining graph by induction with four colors.  If the four nodes are colored with fewer than four colors, then we can use the fourth color to color x.  This is just like we did as in the cases where the degree was 2 or 3.  But if the nodes are colored  with four different colors, B, G, Y, R  then we are dead in the water, because x would require a fifth color E.  For this worst case scenario, Kempe suggested the following clever trick:  Start from the node colored B and create a subgraph by following the edges from this node through all nodes colored B or Y.  Note that the edges between the nodes in this subgraph and the rest of the original graph connect nodes colored B or Y with nodes colored G or R.  This subgraph is known as a Kempe Chain.

Now there are two cases:

    Case 1:  The subgraph does not include the node adjacent to x colored Y.
       This is the easy case.  Kempe instructs us to change the color of all the nodes in the subgraph from B to Y, and from Y to B.  This remains a legal coloring because all edges within the subgraph are still connecting B to Y, and all edges between the subgraph and the original graph are between nodes colored B or Y, and nodes colored G or R.  But look how this helps!  Now x is adjacent to nodes colored Y, G, and R.  Hence x can be colored B and the graph is four-colorable.  See Figure 1.

Note that in all the figures below, the node x is colored white, meaning it has not yet been given a color.


   

    Figure 1 - The Simple Example of a Kempe Chain.  The B and Y nodes on the left side of the graph are toggled, allowing x to be colored B.


    Case 2:  The subgraph includes the node adjacent to x colored YThis is called a cyclical Kempe Chain.  Now Kempe's idea does not work because the two nodes adjacent to x colored B and Y simply switch colors, and x is still adjacent to four different colors.  Kempe fixes the problem by pointing out that if there is a path between the nodes adjacent to x colored B and Y, then there cannot be a path between the nodes colored G and R.  The existence of one cyclical Kempe chain rules out the other, because one acts as a "barrier" between the nodes of the other.  See Figure 2 below to see why.

 

Figure 2 - Cyclical Kempe Chains.  One of the two Kempe chains, either B-Y or R-G, is guaranteed not be cyclical


If there is no path between the nodes colored R or G, then instead of creating the Kempe chain starting from the node colored B, we start it from RThat is, create a subgraph by following the edges from R through all nodes colored R or G.   Then we can toggle the colors R and G in this subgraph, and x ends up adjacent to three colors: B, Y, and G.  See figure 3 below.


                                                                                fig3b
                            

Figure 3  - The Degree 4 Case.  Four different colors become three.

 

Note that so far, Kempe's proof has no flaws.

Degree = 5

Now we have five nodes adjacent to x.  As before, we delete x and color the resulting smaller graph with four colors by induction.  If the five nodes use fewer than four different colors, then we simply color x the fourth color, and we are finished.  If the five adjacent nodes use four different colors, then we must consider two possible cases.  Without any loss of generality, let the nodes adjacent to x be colored B, Y, G, R, and R.  The two cases have to do with the ordering of the nodes in a circle.  See Figures 4 and 5 below. 

Case 1:  The two nodes colored R are next to each other, B-Y-G-R-R (counter-clockwise order). 
Case 2:  The two nodes colored R are separated, B-R-G-Y-R (counter-clockwise order). 



                                                                 

Figure 4 - The Two Cases for Degree 5.  Case 1 is on the eft and Case 2 on the right.

 Case 1 is easy to handle, and the argument is similar to the degree 4 situation.  The details are left as an exercise.  The second case is the climax of Kempe's proof.

Exercise 2.  Complete the proof for Case 1 of Figure 4.


Case 2.  Create a Kempe chain starting from the node colored B, by following all edges that connect to nodes colored B or Y.  As before, if this chain does not contain the node adjacent to x that is colored Y, then we toggle the colors of the chain, resulting in B being colored Y, and that allows us to color x with B.  However, if the chain does contain the node colored Y, then we give up on that chain, and create a new Kempe chain starting from the node colored B, but this time by following all edges that connect to nodes colored B or G.  If this chain does not contain the node adjacent to x that is colored G, then we can toggle the colors of this chain, once again allowing us to color x with B.  If this second chain does contain the node colored G then we must reject this chain as well, and we are left with the hardest case.  We cannot use any Kempe chain starting from B.

Kempe's solution in this case is to start at each of the two nodes colored R and create two Kempe chains, one with colors R and G, and the other with colors R and Y.  From the node colored R that is surrounded by the B-Y Kempe chain (the R on the right), he creates a Kempe chain with colors R and G.  From the node colored R that is surrounded by the B-G Kempe chain (the R on the left), he creates a Kempe chain with colors R and Y.  The new Kempe chain with colors R and G cannot reach the node adjacent to x colored G, so the colors can be toggled, and R becomes G.  The new Kempe chain with colors R and Y cannot reach the node adjacent to x colored Y, so R becomes Y.  Thus the R node on the right is colored G and the R node on the left is colored Y.  This leaves the color R free for x, which is now adjacent to colors Y-G-B-Y-G (in counter-clockwise order around x).  See Figure 5 below.

                                                      
   fig 5



QED


Figure 5 - The Hard Case.  The B-Y and B-G chains are rejected.  Two new chains, one from each R node are used.
The R node on the right is recolored G, and R node on the left is recolored Y, leaving R free for the center node x.

 

Exercise 3.  Note that in Kempe's proof, when x is deleted from the graph, the resulting graph need not be connected.  Look up the definition of strong induction.  Explain why this implies that Kempe's proof uses strong mathematical induction.

The Flaw in Kempe's Proof

The flaw in Kempe's proof, found by Heawood, is in case 2 when the degree of x is 5.  The problem is with the two sentences below.  

"The new Kempe chain with colors R and G cannot reach the node adjacent to x colored G, so the colors can be toggled and R becomes G.  The new Kempe chain with colors R and Y cannot reach the node adjacent to x colored Y, so R becomes Y."

The claim explains correctly why we can color all the nodes adjacent to x without using R, but there is an unexpected problem.  It is possible to have edges between Y nodes in the R-Y Kempe chain and G nodes in the R-G Kempe chain!  See the Y and G nodes at the top of Figure 6. 


                  


Figure 6 - The Flaw

If we change the colors according to Kempe's methods, then both Y and G both get changed to R!   Figure 5 misleads us into thinking that the G nodes surrounded by the B-Y chain cannot be adjacent to any Y nodes surrounded by the B-G chain, but they can!  There it is in Figure 6.  It is helpful to try and convince yourself that this weird situation cannot occur in the degree 4 argument, and that Figure 3 is not similarly misleading.

Here is what John Conway, a famous and flamboyant mathematician, and the inventor of the game Life, has to say about Kempe's proof and the flaw:

"I  did read a few of the papers from this period, including Kempe's proof, Tait's deduction of his edge-coloring criterion from it, and an article in which Heawood pointed out the mistake, among other things, and the impression I got from them was much the same as Jim's. There was indeed a fair amount of interest in Kempe's "theorem", but not much evidence that any great number of people actually scrutinized his proof, or even read it.  I don't think it would have made much difference if they had.  The proof is very convincing, and in the days when the amateur provers were still interested in FCT, 2 times out of 3 "Kempe's Catastrophe", as Tom O'Beirne used to call it, was the proof they'd produce.  Tom  had a set lecture in which he gave Kempe's proof, illustrated by  a specially made-up board with colored pegs, and seldom could anyone  in the audience find anything wrong with it."

A fun paper by Stan Wagon has a colored picture of the famous Errera graph on page 4.   The Errera graph is a complete example of a graph and coloring for which Kempe's proof fails.   One thing to notice is that the flaw requires quite a contrived example, and in practice we may not see this happen very often.  In fact, the same graph but with a different coloring might not reveal the flaw!

Programming Project I


Write a brute force 4-coloring algorithm for planar graphs.  It uses depth first search, backtracking whenever it gets into a situation where none of the four colors work. 

Exercise 4.  The worst case complexity of your algorithm is 4n.  Explain why.


Programming Project II

Create a four coloring algorithm based on Kempe's flawed proof.  Your algorithm may fail due to the flaw in Kempe's proof, but it will always succeed in 5-coloring the graph, just as Kempe's proof is a correct 5-color proof. 

Your algorithm should find a node of degree less than six, delete it from the graph, and recursively color the remainder of the graph.   In the flawed case when the degree = 5, your algorithm will go ahead and build the Kempe chains, as Kempe described in his proof, hoping that it gets lucky and that the  the flaw will not show up.  If the flaw does show up, your algorithm will not be able to continue its four-coloring legally.  You must check to see if this happens, and if it does, you can simply add a fifth color to color the deleted node.   Kempe's flawed method never fails to five-color a graph.


The details of this algorithm are not simple, but they are standard.  You need to use a graph data structure, either an adjacency matrix or an array of adjacency linked lists.  The algorithm works recursively, by searching for any node of degree less than six, and deleting it from the graph.  The resulting graph (or graphs - the pieces  may not all be connected) is/are colored recursively.  You algorithm needs to then color the node it deleted making sure it uses one of the four colors used by the recursive coloring.  The cases are exactly like those in Kempe's proof.  

There is one technical issue that you probably won't think of on your own, that comes up in the case of a deleted node of degree 4 or 5.  Consider the degree 4 case.  When you have the adjacent colors  B, G, Y, and R.   With Kempe's picture, it is clear which nodes are opposite each other.  B is opposite Y, and G is opposite R.  In this case as we described earlier, you  do a depth first search from B looking for colors B and Y, and if necessary, another depth first search from R looking for colors R and G.  In the end, you  have a free color for the deleted node that you are trying to color.  However, in a regular graph data structure it is not at all obvious which nodes are "opposite" each other.  That is, we have no way to know which nodes share a face, because we have no special data structure for the details of the planar embedding.  Hence if you started a depth first search from a node with a particular color, you would not know which other color to search for.   

The easiest way to handle this without opening a Pandora's box of data structures, is to 

  1. Do not search from a node toward a node that is adjacent to it.  Such nodes could not possible be "opposite" as shown in the figures.
  2. If all 3 of the available nodes are not adjacent, then try each of them one at a time.  When you try the wrong pairs of nodes, it is possible to find two cyclical Kempe chains.  In this case, the recoloring does not free up a color for the deleted node.  If this happens, just try the next node.  Kempe's proof implies that there is a correct choice - that is one where two cyclical Kempe chains are impossible.  Therefore if you try all 3 possibilities, you will be able to color the deleted node.

The case for degree 5 is similar, and you will need to try all 4 possible nodes as opposite.  Of course the flaw is in the degree 5 case, so if none of them of the 4 possibilities work, you add a fifth color as described earlier.

You should experiment with your algorithm and see if it ever fails.  Use the graph of the map of the USA which you can find here.

Exercise 5.  What is the complexity of your Kempe-based algorithm?  Explain your analysis.

Note, there is a quadratic time algorithm to 4-color a planar graph, but it uses techniques from a correct proof that are too hard for us to consider here.  Is your pseudo-correct algorithm faster or slower than this one?