7 Traversing Graphs
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\(\def\omega{\unicode{x1D714}}\)
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\(\def\itNu{\unicode{x1D6EE}}\)
\(\def\itXi{\unicode{x1D6EF}}\)
\(\def\itOmicron{\unicode{x1D6F0}}\)
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7.5 Chapter Notes¶
The emergence of graph theory as a mathematical discipline is closely linked to a puzzle called the seven bridges of Königsberg . In the 18th century, the city of Königsberg (now Kaliningrad) was divided by the river Pregel into four districts connected by seven bridges (Fig. 7.6 ).
In the figure, the districts are labeled with upper-case letter A to D and the bridges with lower-case letter a to g . The question is: Is there a tour of the city that starts and ends in the same district and crosses all seven bridges exactly once?
Figure 7.6 Illustration of the Seven Bridges of Königsberg problem.
In this particular instance, it’s not hard to try all possible routes and conclude that no such tour exists, but what about more general versions of the puzzle, with different arrangements of districts and bridges? In 1736, the German mathematician Leonard Euler proved that a cyclic tour that crosses every
bridge exactly once can only exist if every district has an even number of bridges [33 ]. This result is considered the first major theorem in the field we now refer to as graph theory : It is known as Euler’s theorem , and a cyclic path through a graph that contains every edge exactly once is called an Eulerian cycle .
Another famous puzzle that was settled with the help of graph theory is the four color problem . The problem was first posed in the early 1850s and asks whether it is possible to color any given map in such a way that only four colors are used and no two countries with a shared border have the same color. A formal proof
that this is indeed possible eluded mathematicians for more than a century and was only found in 1976 by Kenneth Appel and Wolfgang Haken [4 , 5 ]. In
their proof, Appel and Haken first reduced the infinite number of possible maps to 1834 special graphs, and then used a computer program to verify that each of these graphs is in fact 4-colorable. Appel and Haken’s proof was one of the first mathematical proofs that heavily relied on computer
programming, something that irked many mathematicians at the time. To this day, no proof of the four-color theorem has been found that doesn’t require the use of computers.
In this chapter we have discussed how to model and solve several different puzzles with the help of graphs and graph algorithms . Graph algorithms are discussed in more detail in most textbooks on algorithms and data structures [86 , 25 ]. If you are interested in the mathematical theory of graphs , the book by Benjamin et al. [14 ] is an excellent introduction to the subject.
For a more practical take on graph algorithms, see The Stanford GraphBase by Donald E. Knuth [63 ], which discusses the implementation of a wide range of graph algorithms and their applications to puzzles and other
combinatorial problems. An extensive collection of algorithmic puzzles can be found in the book by Levitin and Levitin [66 ].
The wolf-goat-cabbage problem in Exercise 7.6 and the jealous husbands problem in Exercise 7.7 belong to the class of river-crossing puzzles in which a certain number of people or other objects must cross one or more rivers according to several constraints. River-crossing puzzles first appear in one of the earliest known collection of mathematical problems, a
book called Propositiones ad Acuendos Juvenes (“Problems to Sharpen the Young”) that is believed to have been written around 800 AD [46 ].
Many river-crossing puzzles have a hidden structure that can be studied mathematically. For instance, it is possible to prove that the \(n\)-jealous-couples problem cannot has a solution for \(n\ge 4\) only if there is an island, and that in this case the best solution requires \(4n+1\) crossings; for details, see
the article by Pressman and Singmaster [78 ].
The die-hard problem we discussed at the beginning of this chapter was inspired by the movie of the same name, but puzzles of this kind have a much longer history and are known as water measuring problems or decanting problems . The first written record of such puzzles can be found in the Annales Stadenses , a 13th century world chronicle written by the German monk Albert of Stade. The original version of the problem differs slightly from the one we discussed at the
start of this chapter: Given a jug that contains 8 units of wine and two additional jugs that can hold 5 and 3 units, respectively, is it possible to divide the wine equally? (We discuss this three-jug problem in Exercise 7.8 .)
A clever graphical method for solving this kind of problem was discovered by M. C. K. Tweedie in 1939 and later popularized by Martin Gardner [93 , 39 ]. An illustration of this method is shown in Fig. 7.7 . The main idea is to draw the die-hard graph as a parallelogram, with the four corners again representing the
states in which each jug is either full or empty. When visualized like this, the shortest path from \((0,0)\) to \((4,3)\) resembles the path of a billiard ball that starts in the lower-left corner and then repeatedly bounces off the sides of the parallelogram. The method
can be generalized to arbitrary two-jug and three-jug problems; for details, see the references cited above.
Figure 7.7 Solving the die-hard problem using bouncing balls.