MT4514: Graph Theory

Chapter 1Introduction1. About the courseThe course will not be solely based on a single book. Therefore, the best studysource will be the lecture notes. Some useful texts are:1. Robin Wilson, Introduction to Graph Theory2. Robin Wilson and John Watkins, Graphs – an Introductory Approach.3. Frank Harary, Graph Theory.4. Norman Biggs, Discrete MathematicsAll these books, as well as all tutorial sheets and solutions, will be available inMathematics/Physics library on short loan. Also, any other book containing in itstitle the words such as ‘graph theory’, ‘discrete mathematics’, ‘combinatorics’ islikely to contain material relevant to the course.It will be useful to bring coloured pens or pencils to lectures, although I’ve hadto do these notes in black and white.2. Some introductory examplesGraphs are a very useful way of recording information about objects and relationshipsbetween pairs of objects. A graph consists of a set V of vertices or points anda collection E of edges joining some of the pairs of points.In this introduction we’ll look at a range of examples to indicate the flavour andvariety of the topic. Graph theory goes back to ancient Greek times, with the studyof the 5 regular platonic solids, but it really started with the following problem.Example 2.1. Königsberg Bridge problem. Consider this town plan of Königsberg(now Kaliningrad) in East Prussia. The river Pregel runs through the middle oftown, and a set of 7 bridges links the banks and two islands in the river. Theproblem is to find a walk through the town that begins and ends at the same place,crossing each bridge exactly once.3