MTH 103: ontemporary Mathematics Study Guide for hapter 5: Euler ircuits 1. Use the graphs below to practice the following: (a) etermine which ones have Euler ircuits, and which ones have Euler Paths. (b) For the graphs which have an Euler ircuit or an Euler Path, use Fleury s lgorithm to find such ircuit or Path. O W P Y R Q V X Y S T U Z W G M N O F E R Q P 2. Read Sect. 5.1 and 5.2. Pay particular attention to the notation. We defined many terms in class, use your class notes to review them and then read the book, which gives careful definition of each term. Write the meaning of the following words: vertex: edge: loop: adjacent vertices: path: connected graph: components of a graph: Euler path: degree of a vertex: circuit: disconnected graph: bridge: Euler circuit:
3. Try problems 5, 8, 11 at page 186 4. The most important stepin constructing a graph model of a certain situation is to identify two types of elements: objects and relationships between pairs of objects. Objects are modelled by vertices, while relationships are modelled by edges. Practice problems 15, 17, 19, 20 on page 187, focusing the construction of graph models for different situations. 5. Explain the difference between a path and a circuit. 6. What determines whether a given circuit in a graph is an Euler circuit? 7. You will now be asked to give examples of graphs with certain properties. You can look at the examples in your book and notes, but you must come up with your own graph (be creative!). (Note that every edge must terminate in TWO vertices) (a) Give an example of a graph in which all the vertices have even degree. (b) Give an example of disconnected graph. (c) Give an example of a graph with at least one bridge. Indicate a bridge by highlighting it. (d) Give an example of a graph in which no edge is a bridge. (e) raw a graph in which every edge is a bridge. (f) Use Euler s results to provide an example of a graph which has no Euler circuit.
(g) Read Euler s First Theorem and then provide an example of a graph which has an Euler path, but no Euler circuit. (h) Read Euler s Second Theorem and then provide an example of a graph which has no Euler circuit and no Euler path. 8. State here the famous (and useful!) Handshaking Theorem. o you remember what reasoning we used to justify it? Review class notes for our discussion and then do problems 45, 53, 54 (somewhat challenging). 9. If the Handshaking Theorem is clear, complete the following sentences: (a) graph with 33 edges has a total degree of (b) In a graph with 94 vertices of which TWO vertices have degree 2 and all others have degree 1, there are edges. (c) Statement: Each of the 7 vertices of a given graph has odd degree. This statement is (Justify) 10. Practice problems 37-41 p 190. When the book talks about unicursal tracing, it simply means tracing the line drawing without skipping or retracing lines, that means finding an Euler circuit or an Euler path in the graph. 11. Use Fleury s algorithm to find an Euler circuit for the graph below. You may demonstrate that you found an Euler s circuit by either (a) numbering the edges in the order in which you travel along them, or (b) listing the vertices in the order in which you travel through them. E 00000000000000000000 11 01 1100 01 0011 F 0011 01 11 000 11 0 1 01 01 01 01 01 0 1 01 01 010 1 01 01 01 01 01 0011 01 01 0011 01 01 00 00 00 11 01 0011 01 0011 01 01 01 0011 01 01 0 1 01 1100 1 01 0 100 11 01 01 G 0 1 0 1
12. In Fleury s algorithm for finding an Euler circuit or Euler path, we do not travel an edge which is a bridge for the untravelled portion of the graph. (a) What would happen if we travelled such an edge? Why do we avoid travelling such an edge? (b) If we travelled such an edge, would we be cut off from the starting vertex? (careful, the correct answer requires some thinking and maybe an example). Explain carefully. (c) Think about the difference in the phrases bridge for the graph and bridge for the untravelled portion of the graph. The word untravelled in the second phrase is very important in Fleury s algorithm. 13. Euler s results discuss graphs which have two or more odd degree vertices. (a) What about graphs with no vertices of odd degree? o they have Euler paths? (b) What about graphs with exactly one odd degree vertex? Is there any such graph? If so, give an example. If not, explain why. 14. Practice Eulerization. Remember that when you duplicate existing edges, duplication must be interpreted as travelling twice along an edge of the graph given initially. You will find carefully worked out examples of Eulerization on page 180. Good problems to practice are 41 and 42 on p. 190. 15. n Eulerization is called optimal if it is achieved with the least number of duplications. (a) Suppose a given graph has some vertices of odd degree and some of even degree. We count 24 odd degree vertices and we proceed to find its Eulerization. If, to do that, we have to duplicate 12 edges, is our Eulerization optimal? Explain. (b) Produce two different Eulerizations of the graph from problem 44 (a): one optimal Eulerization and one non-optimal Eulerization. Make sure you understand the difference between the two.
16. raw the graph appearing in problem 44 (a) p. 190 again. uplicate enough edges so that all of the vertices UT 2 are even. This procedure is called semi-eulerization and produces a graph which has an Euler path (it provides models for those problems in which the preferred route does not need to return to the start vertex) 17. raw two copies of the graph in problem 42 (a) p. 188 below: (a) Find an optimal semi-eulerization of this graph. Show your work on one of the copies. (b) Label each vertex of the semi-eulerized graph. Then find an Euler path using Fleury s algorithm. Write the Euler path below as a list of vertices. (c) The Euler path you found is one that retraces the least number of edges of the original graph. Make sure you understand this interpretation. (d) How many edges will be retraced in this optimal route? (e) If the original graph represents a city neighborhood and it takes a police patrol 10 minutes to walk along each street block (i.e. each edge of the graph), how long will it take him to walk an optimal route?